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On the Canonical Formulation of Electrodynamics and Wave Mechanics


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ON THE CANONICAL F ORMULA TION OF ELECTR OD YNAMICS AND W A VE MECHANICS By D A VID JOHN MASIELLO A DISSER T A TION PRESENTED TO THE GRADUA TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORID A 2004

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F or Katie.

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A CKNO WLEDGMENTS Since August of 1999, I ha v e had the privilege of conducting m y Ph.D. researc h in the group of Prof. Yngv e Ohrn and Dr. Erik Deumens at the Univ ersit y of Florida's Quan tum Theory Pro ject. During m y time in their group I learned a great deal on the theory of dynamics, in particular, the Hamiltonian approac h to dynamics and its applications in electro dynamics and atomic and molecular collisions. I also learned a new appreciation for scien tic computing, of whic h I w as previously ignoran t. Most imp ortan tly Prof. Ohrn and Dr. Deumens taugh t me ho w to think through a ph ysical problem, sort out its underlying dynamical equations, and solv e them in a mathematically w ell-dened manner. I esp ecially w an t to thank Dr. Erik Deumens, with whom I w ork ed most closely during m y Ph.D. researc h. Erik had a vision when I b egan m y graduate studies and has promoted m y w ork since then to successfully realize it. Along the w a y he c hallenged m y creativ e, mathematical, and ph ysical in tuitions and imparted on me a lo v e for theoretical ph ysics. Erik has alw a ys tak en time to listen to and carefully answ er m y questions and has alw a ys resp ected m y ideas. I thank him for b eing suc h an excellen t men tor to me. My understanding of ph ysics has also b een broadened b y man y others. Firstly I w ould lik e to thank Dr. Remigio Cabrera-T rujillo, who w as a p ost do ctoral asso ciate in the Ohrn-Deumens group, for his guidance esp ecially during m y rst few y ears. He has b een a great source for advice on man y topics from the details of quan tum scattering theory to simple computer problems lik e clearing prin ter jams. I ha v e jok ed on man y o ccasions that he w as m y p ersonal p ostdo c b ecause he w as alw a ys so willing to help when I had questions. I w ould also lik e to thank the past and presen t mem b ers of m y researc h group, in particular, Dr. Anatol Blass, Dr. Maur cio iii

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Coutinho Neto, Mr. Ben Killian, and Mr. Virg F ermo. In addition, I w ould lik e to thank m y ocemates with whom I ha v e sp en t almost v e y ears. I thank Ms. Ariana Beste, Mr. Igor Sc h w eigert, and Mr. T om Henderson for their friendship and camaraderie. I ha v e esp ecially b eneted from man y con v ersations with T om Henderson on asp ects of quan tum mec hanics, quan tum eld theory and classical electro dynamics. Sev eral other facult y and sta at the Quan tum Theory Pro ject, and the Departmen ts of Chemistry Ph ysics, and Mathematics at the Univ ersit y of Florida ha v e also encouraged and promoted m y Ph.D. researc h. A t the Quan tum Theory Pro ject, I thank Prof. Je Krause for taking sincere in terest in m y researc h and alw a ys nding time to listen to me and pro vide guidance. I ha v e taugh t with Je on a few o ccasions and ha v e kno wn him to b e a great teac her as w ell as men tor. I thank Prof. Henk Monkhorst for his kindness and go o d h umor. I will esp ecially miss all of the L A T E X battles that w e ha v e fough t o v er the past sev eral y ears. In addition, I w ould lik e to thank Dr. Ajith P erera for his friendship and patience. I thank the sta, esp ecially Ms. Judy P ark er and Ms. Coralu Clemen ts, for k eeping all of the administrativ e asp ects of m y graduate studies running smo othly I w ould also lik e to thank the custo dians Sandra and Rhonda who ha v e b een so friendly to me and who k eep the Quan tum Theory Pro ject imp eccably clean. In the Departmen t of Chemistry I w ould lik e to thank the late Prof. Carl Stoufer, who w as m y undergraduate advisor during m y rst y ear, for his friendship, wisdom, and advice. Throughout m y en tire undergraduate career w e w ould meet a few times p er y ear to catc h up o v er coee and don uts. It w as due to Carl's supp ort that I w as giv en the opp ortunit y to study at the Quan tum Theory Pro ject. In the Departmen t of Ph ysics, I w ould lik e to thank Prof. Ric hard W o o dard, from whom I learned quan tum eld theory Ric hard is v ery passionate ab out ph ysics and is p erhaps the b est teac her that I ha v e kno wn. F rom him I gained a deep er understanding of p erturbation theory and its applications in iv

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quan tum electro dynamics. In the Departmen t of Mathematics, I w ould lik e to thank Prof. Scott McCullough, who w as eectiv ely m y undergraduate advisor. While I w as an undergraduate studen t of Scott's, he imparted to me a deep appreciation for mathematics and a particular in terest in analysis. Scott w as an excellen t teac her and men tor, and under his guidance, m y undergraduate researc h w as a w arded b y the College of Lib eral Arts and Sciences. Outside of the Univ ersit y of Florida, man y others ha v e con tributed to m y scientic career. A t the Univ ersit y of Cen tral Florida's Cen ter for Researc h and Education in Optics and Lasers, I w ould lik e to thank Prof. Leonid Gleb o v, Prof. Kathleen Ric hardson, and Prof. Boris Zel'do vic h for rst in tro ducing me to the w orld of quantum ph ysics. In particular, Prof. Gleb o v and Prof. Ric hardson greatly stim ulated and encouraged m y in terests. With their recommendation, I receiv ed a fello wship to study at the Univ ersit y of Bordeaux's Departmen t of Ph ysics and Cen tre de Ph ysique Mol eculaire Optique et Herzienne. While in Bordeaux, F rance, I had the pleasure of w orking in the researc h group of Prof. Lauren t Sarger. I wish to thank Prof. Sarger as w ell as his colleagues for their hospitalit y during m y time in F rance and for in tro ducing me to the eld of atomic and molecular ph ysics, whic h is the setting for this dissertation. Lastly I w ould lik e to thank m y family My mother and father ha v e alw a ys pro vided unconditional lo v e, supp ort, and guidance to me. They ha v e encouraged m y inquisitiv eness of Nature and ha v e promoted m y education from kindergarten to Ph.D. I thank m y inla ws for their lo v e and supp ort and for pro viding a home a w a y from home while in graduate sc ho ol. In conclusion, I w ould lik e to thank m y w onderful wife Katie for her encouragemen t, companionship, and unending lo v e. v

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T ABLE OF CONTENTS pageA CKNO WLEDGMENTS. . . . . . . . . . . . . . . iiiLIST OF FIGURES. . . . . . . . . . . . . . . . ixABSTRA CT. . . . . . . . . . . . . . . . . . xiCHAPTER 1 INTR ODUCTION. . . . . . . . . . . . . . . 11.1 Ph ysical Motiv ation. . . . . . . . . . . . . 11.2 Historical and Mathematical Bac kground. . . . . . . 21.2.1 Gauge Symmetry of Electro dynamics. . . . . . 31.2.2 Gauge Symmetry of Electro dynamics and W a v e Mec hanics51.3 Approac hes to the Solution of the Maxw ell-Sc hr odinger Equations71.4 Canonical F orm ulation of the Maxw ell-Sc hr odinger Equations. . 111.5 F ormat of Dissertation. . . . . . . . . . . . 141.6 Notation and Units. . . . . . . . . . . . . 152 THE D YNAMICS. . . . . . . . . . . . . . . 172.1 Lagrangian F ormalism. . . . . . . . . . . . 182.1.1 Hamilton's Principle. . . . . . . . . . . 192.1.2 Example: The Harmonic Oscillator in ( q k ; q k ). . . . 212.1.3 Geometry of T Q. . . . . . . . . . . . 222.2 Hamiltonian F ormalism. . . . . . . . . . . . 242.2.1 Example: The Harmonic Oscillator in ( q a ; p a ). . . . 252.2.2 Symplectic Structure and P oisson Brac k ets. . . . . 262.2.3 Geometry of T Q. . . . . . . . . . . . 273 ELECTR OD YNAMICS AND QUANTUM MECHANICS. . . . 293.1 Quan tum Mec hanics in the Presence of an Electromagnetic Field. 293.1.1 Time-Dep enden t P erturbation Theory. . . . . . 303.1.2 F ermi Golden Rule. . . . . . . . . . . 333.1.3 Absorption of Electromagnetic Radiation b y an A tom. . 343.1.4 Quan tum Electro dynamics in Brief. . . . . . . 363.2 Classical Electro dynamics Sp ecifed b y the Sources and J. . 403.2.1 Electromagnetic Radiation from an Oscillating Source. . 413.2.2 Electromagnetic Radiation from a Gaussian W a v epac k et. 47 vi

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4 CANONICAL STR UCTURE. . . . . . . . . . . . 554.1 Lagrangian Electro dynamics. . . . . . . . . . . 564.1.1 Cho osing a Gauge. . . . . . . . . . . . 564.1.2 The Lorenz and Coulom b Gauges. . . . . . . 574.2 Hamiltonian Electro dynamics. . . . . . . . . . 594.2.1 Hamiltonian F orm ulation of the Lorenz Gauge. . . . 614.2.2 P oisson Brac k et for Electro dynamics. . . . . . . 664.3 Hamiltonian Electro dynamics and W a v e Mec hanics in Complex Phase Space. . . . . . . . . . . . . . . 664.4 Hamiltonian Electro dynamics and W a v e Mec hanics in Real Phase Space. . . . . . . . . . . . . . . . 694.5 The Coulom b Reference b y Canonical T ransformation. . . . 704.5.1 Symplectic T ransformation to the Coulom b Reference. . 714.5.2 The Coulom b Reference b y Change of V ariable. . . . 784.6 Electron Spin in the P auli Theory. . . . . . . . . 794.7 Proton Dynamics. . . . . . . . . . . . . . 815 NUMERICAL IMPLEMENT A TION. . . . . . . . . . 845.1 Maxw ell-Sc hr odinger Theory in a Complex Basis. . . . . 855.2 Maxw ell-Sc hr odinger Theory in a Real Basis. . . . . . 885.2.1 Ov erview of Computer Program. . . . . . . . 905.2.2 Stationary States: s and p -W a v es. . . . . . . 935.2.3 Nonstationary State: Mixture of s and p -W a v es. . . 935.2.4 F ree Electro dynamics. . . . . . . . . . . 935.2.5 Analysis of Solutions in Numerical Basis. . . . . 955.3 Symplectic T ransformation to the Coulom b Reference. . . . 995.3.1 Numerical Implemen tation. . . . . . . . . 1015.3.2 Stationary States: s and p -W a v es. . . . . . . 1025.3.3 Nonstationary State: Mixture of s and p -W a v es. . . 1025.3.4 F ree Electro dynamics. . . . . . . . . . . 1035.3.5 Analysis of Solutions in Coulom b Basis. . . . . . 1035.4 Asymptotic Radiation. . . . . . . . . . . . 1035.5 Proton Dynamics in a Real Basis. . . . . . . . . 1086 CONCLUSION. . . . . . . . . . . . . . . . 110APPENDIX A GA UGE TRANSF ORMA TIONS. . . . . . . . . . . 113A.1 Gauge Symmetry of Electro dynamics. . . . . . . . 113A.2 Gauge Symmetry of Quan tum Mec hanics. . . . . . . 115 vii

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B GREEN'S FUNCTIONS . . . . . . . . . . . . . 116 B.1 The Dirac -F unction . . . . . . . . . . . . 117 B.2 The r 2 Op erator . . . . . . . . . . . . . 118 B.3 The @ 2 Op erator . . . . . . . . . . . . . 118 C THE TRANSVERSE PR OJECTION OF A ( x ; t ) . . . . . . 122 C.1 T ensor Calculus . . . . . . . . . . . . . . 125 C.2 T 0 k k ( x 0 ; t ) In tegrals . . . . . . . . . . . . . 127 C.2.1 Inside Step . . . . . . . . . . . . . 129 C.2.2 Outside Step . . . . . . . . . . . . . 131 C.3 Building A T ( x ; t ) . . . . . . . . . . . . . 133 REFERENCES . . . . . . . . . . . . . . . . . 134 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . 142 viii

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LIST OF FIGURES Figure page 2{1 The conguration manifold Q = S 2 is depicted together with the tangen t plane T q k Q at the p oin t q k 2 Q : . . . . . . . 22 3{1 The co ecien t 1 of the unscattered plane w a v e exp ( i k x ) is analogous to the 1 part of the S -matrix, while the scattering amplitude f k (n) whic h mo dulates the scattered spherical w a v e exp ( ik r ) =r is analogous to the iT part. . . . . . . . . . . . 38 3{2 In the radiation zone, the observ ation p oin t x is lo cated far from the source J : In this case the distance j x x 0 j r ^ n x 0 : . . . 44 3{3 The dieren tial p o w er dP =d n or radiation pattern corresp onding to an oscillating electric dip ole v eries that no radiation is emitted in the direction of the dip ole momen t. . . . . . . . . . 46 3{4 The norms of J and A are plotted with dieren t v elo cities along the x -axis. . . . . . . . . . . . . . . . . . 48 3{5 The tra jectory or w orld line r ( t ) of the c harge is plotted. . . . 49 3{6 The bremsstrahlung radiation from a c harged gaussian w a v epac k et mo v es out on the smeared ligh t cone with maxim um at x = ct: . 50 3{7 The radiation pattern giv en b y ( 3.63 ) sho ws the c haracteristic dip ole pattern at lo w est order. . . . . . . . . . . . . 53 4{1 A limited but relev an t p ortion of the gauge story in the Lagrangian formalism is organized in this picture. . . . . . . . . 59 4{2 The Hamiltonian form ulation of the gauge story is organized in this picture with resp ect to the previous Lagrangian form ulation. . . 65 4{3 Comm utivit y diagram represen ting the c hange of co ordinates ( q ; p ) to ( ~ p; ~ q ) at b oth the Lagrangian and equation of motion lev els. . . 79 5{1 Sc hematic o v erview of ENRD computer program. . . . . . 92 5{2 Phase space con tour for the co ecien ts of the v ector p oten tial A and its momen tum : . . . . . . . . . . . . . . 94 5{3 Phase space con tour for the co ecien ts of the real-v alued Sc hr odinger eld Q and its momen tum P : . . . . . . . . . . 94 ix

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5{4 Phase space con tour for the co ecien ts of the scalar p oten tial and its momen tum : . . . . . . . . . . . . . . 95 5{5 Real part of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es. . . . . . . . 97 5{6 Imaginary part of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es. . . . . . 97 5{7 Probabilit y for the electron to b e in a particular basis eigenstate. . 98 5{8 Phase space of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es. . . . . . . . 98 5{9 Real part of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es. . . . . . . . 104 5{10 Imaginary part of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es. . . . . . 104 5{11 Probabilit y for the electron to b e in a particular basis eigenstate. . 105 5{12 Phase space of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es. . . . . . . . 105 5{13 Sc hematic picture of the lo cal and asymptotic basis prop osed for the description of electromagnetic radiation and electron ionization. . 107 B{1 The tra jectory or w orld line r ( t ) of a massiv e particle mo v es from past to future within the ligh t cone. . . . . . . . . . . 120 C{1 Since ~ A = ~ h v ; the transv erse v ector p oten tial ~ A ? = [ v k ( k v ) =k 2 ] ~ h and the longitudinal v ector p oten tial ~ A k = [ k ( k v ) =k 2 ] ~ h; where ~ h is a scalar function. . . . . . . . . . . . . . 122 x

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Abstract of Dissertation Presen ted to the Graduate Sc ho ol of the Univ ersit y of Florida in P artial F ulllmen t of the Requiremen ts for the Degree of Do ctor of Philosoph y ON THE CANONICAL F ORMULA TION OF ELECTR OD YNAMICS AND W A VE MECHANICS By Da vid John Masiello Ma y 2004 Chair: Nils Yngv e Ohrn Ma jor Departmen t: Chemistry The in teraction of electromagnetic radiation with atoms or molecules is often understo o d when the timescale for the electromagnetic deca y of an excited state is separated b y orders of magnitude from the timescale of the excited state's dynamics. In these cases, the t w o dynamics ma y b e treated separately and a p erturbativ e F ermi golden rule analysis is appropriate. Ho w ev er, there do exist situations where the dynamics of the electromagnetic eld and the atomic or molecular system o ccurs on the same timescale, e.g. photon-exciton dynamics in conjugated p olymers and atom-photon dynamics in cold atom collisions. Nonp erturbativ e metho ds for the solution of the coupled nonlinear Maxw ellSc hr odinger dieren tial equations are dev elop ed in this dissertation whic h allo w for the atomic or molecular and electromagnetic dynamics to o ccur on the same timescale. These equations ha v e b een deriv ed within the Hamiltonian or canonical formalism. The canonical approac h to dynamics, whic h b egins with the Maxw ell and Sc hr odinger Lagrangians together with a Lorenz gauge xing term, yields a set of rst order Hamilton equations whic h form a w ell-p osed initial v alue problem. That is, their solution is uniquely determined and kno wn in principle once xi

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the initial v alues for eac h of the asso ciated dynamical v ariables are sp ecied. The equations are also closed since the Sc hr odinger w a v efunction is c hosen to b e the source for the electromagnetic eld and the electromagnetic eld reacts bac k up on the w a v efunction. In practice, the Maxw ell-Sc hr odinger Lagrangian is represen ted in a basis of gaussian functions with dieren t widths and cen ters. Application of the calculus of v ariations leads to a set of Euler-Lagrange equations that, for that c hoice of basis, form and represen t the coupled rst order Maxw ell-Sc hr odinger equations. In the limit of a complete basis these equations are exact and for an y nite c hoice of basis they pro vide an appro ximate system of dynamical equations that can b e in tegrated in time and made systematically more accurate b y enric hing the basis. These equations are n umerically implemen ted for a basis of arbitrary nite rank. The dynamics of the basis-represen ted Maxw ell-Sc hr odinger system is in v estigated for the spinless h ydrogen atom in teracting with the electromagnetic eld. xii

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CHAPTER 1 INTR ODUCTION Chemistry encompasses a broad range of Nature that v aries o v er orders of magnitude in energy from the ultracold nK Bose-Einstein condensation temp eratures [ 1 2 ] to the k eV collision energies that pro duce the Earth's aurorae [ 3 { 5 ]. A t the most fundamen tal lev el, the study of c hemistry is the study of electrons and n uclei. The in teraction of electrons and n uclei throughout this energy regime is mediated b y the photon whic h is the quan tum of the electromagnetic eld. The equations whic h go v ern the dynamics of electrons, n uclei, and photons are therefore the same equations whic h go v ern all of c hemistry [ 6 ]. They are the Sc hr odinger equation [ 7 8 ] i = H (1.1) and Maxw ell's equations [ 9 ] r E = 4 r B = 4 c J + E c r B = 0 r E + B c = 0 : (1.2) As they stand these equations are uncoupled. The solutions of the Sc hr odinger equation ( 1.1 ) do not a priori inruence the solutions of the Maxw ell equations ( 1.2 ) and vice v ersa. The dev elopmen t of analytic and n umerical metho ds for the solution of the coupled Maxw ell-Sc hr odinger equations is the main purp ose of this dissertation. Before delving in to the details of these metho ds a ph ysical motiv ation as w ell as a historical and mathematical bac kground is pro vided. 1.1 Ph ysical Motiv ation Man y situations of ph ysical in terest are describ ed b y the system of Maxw ellSc hr odinger equations. Often these situations in v olv e electromagnetic pro cesses that o ccur on drastically dieren t timescales from that of the matter. An example of suc h 1

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2 a situation is the stim ulated absorption or emission of electromagnetic radiation b y a molecule. The description of this pro cess b y ( 1.1 ) and ( 1.2 ) accoun ts for a theoretical understanding of all of sp ectroscop y whic h has pro vided an immense b o dy of c hemical kno wledge. Ho w ev er, there do exist situations where the dynamics of the electromagnetic eld and the matter o ccur on the same timescale. F or example, in solid state ph ysics certain electronic w a v epac k ets exp osed to strong magnetic elds in semiconductor quan tum w ells are predicted to demonstrate rapid decoherence [ 10 ]. The dynamics of the inciden t eld, the electronic w a v epac k et, and the phonons that it emits is coupled and o ccurs on the same fem tosecond timescale. In atomic ph ysics, the long timescale for the dynamics of cold and ultracold collisions of atoms in electromagnetic traps has b een observ ed to exceed lifetimes of excited states, whic h are on the order of 10 8 s. This means that sp on taneous emission can o ccur during the course of collision and ma y signican tly alter the atomic collision dynamics [ 11 12 ]. Cold atom phenomena are also b eing merged with ca vit y quan tum electro dynamics to realize single atom lasers [ 13 { 15 ]. The function of these no v el devices is based on strong coupling of the atom to a single mo de of the resonan t ca vit y Lastly in p olymer c hemistry ultrafast ligh t emission has b een detected in certain ladder p olymer lms follo wing ultrafast laser excitation [ 16 ]. A fundamen tal understanding of the w a v eguiding pro cess that o ccurs in these p olymers is unkno wn. It is pr e cisely these situations, wher e the ele ctr omagnetic and matter dynamics o c cur on the same timesc ale and ar e str ongly c ouple d, that ar e the motivation for this dissertation. 1.2 Historical and Mathematical Bac kground The history of the Maxw ell-Sc hr odinger equations dates bac k to the early t w entieth cen tury when the founding fathers of quan tum mec hanics w ork ed out the theoretical details of the in teraction of electro dynamics with quan tum mec hanics [ 17 ]. It w as realized early on that the electromagnetic coupling to matter w as through

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3 the p oten tials and A ; and not the elds E and B themselv es [ 6 18 19 ]. The p oten tials and elds are related b y E = r A =c B = r A (1.3) whic h can b e conrmed b y insp ecting the homogeneous Maxw ell equations in ( 1.2 ). Unlik e in classical theory where the p oten tials w ere in tro duced as a con v enien t mathematical to ol, the quan tum theory requires the p oten tials and not the elds. That is, the p oten tials are fundamen tal dynamical v ariables of the quan tum theory but the elds are not. A concrete demonstration of this fact w as presen ted in 1959 b y Aharono v and Bohm [ 20 ]. 1.2.1 Gauge Symmetry of Electro dynamics It w as w ell kno wn from the classical theory of electro dynamics [ 9 ] that w orking with the p oten tials leads to a p oten tial form of Maxw ell's equations that is more rexible than that in terms of the elds alone ( 1.2 ). In p oten tial form, Maxw ell's equations b ecome r 2 A A c 2 r h r A + c i = 4 c J (1.4a) r 2 + r A c = 4 : (1.4b) The homogeneous Maxw ell equations are iden tically satised. These p oten tial equations enjo y a symmetry that is not presen t in the eld equations ( 1.2 ). This symmetry is called the gauge symmetry and can b e generated b y the transformation A A 0 = A + r F 0 = F =c; (1.5) where F is a w ell-b eha v ed but otherwise arbitrary function called the gauge generator. Applying this gauge tr ansformation to the p oten tials in ( 1.4 ) leads to exactly the same set of p oten tial equations. In other w ords, these equations are in v arian t under arbitrary gauge transformation or are gauge invariant. They p ossess the

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4 full gauge symmetry Notice also that the electric and magnetic elds are gauge in v arian t. In fact, it turns out that all ph ysical observ ables are gauge in v arian t. That electro dynamics p ossesses gauge symmetry places it in a league of theories kno wn as gauge theories [ 21 ]. These theories include general relativit y [ 22 23 ] and Y ang-Mills theory [ 24 { 26 ]. Gauge theories all suer from an indeterminateness due to their gauge symmetry In an eort to deal with this indeterminateness, it is common to rst eliminate the symmetry (usually up to the residual symmetry; see Chapter 4 ) b y gauge xing and then w ork within that particular gauge. That is, the rexibilit y implied b y the gauge transformation ( 1.5 ) allo ws for the p oten tials to satisfy certain constrain ts. These constrain ts imply a particular c hoice of gauge and gauge generator. Gauge xing is the act of constraining the p oten tials to satisfy a certain constrain t throughout space-time. F or example, in electro dynamics the p oten tial equations ( 1.4 ) r 2 A A c 2 r h r A + c i = 4 c J r 2 + r A c = 4 ( 1.4 ) form an ill-p osed initial v alue problem. Ho w ev er, they can b e con v erted to a w elldened initial v alue problem b y adding an equation of constrain t to them. F or example, adding the constrain t =c + r A = 0 leads to the w ell-dened Lorenz gauge equations r 2 A A c 2 = 4 c J r 2 c 2 = 4 (1.6) while adding r A = 0 leads to the w ell-dened Coulom b gauge equations r 2 A A c 2 = 4 c J T r 2 = 4 ; (1.7) where J T is the transv erse pro jection of the curren t J (see App endix A ). There are man y other c hoices of constrain t, eac h leading to a dieren t gauge. It is alw a ys

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5 p ossible to nd a gauge function that will transform an arbitrary set of p oten tials to satisfy a particular gauge constrain t. The sub ject of the gauge symmetry of electro dynamics, whic h is a subtle but fundamen tal asp ect of this dissertation, is discussed in detail in Chapter 4 In particular, it will b e argued that xing a particular gauge, whic h in turn eliminates the gauge from the theory is not necessarily optimal. Rather, it is stressed that the gauge freedom is a fundamen tal v ariable of the theory and has its o wn dynamics. 1.2.2 Gauge Symmetry of Electro dynamics and W a v e Mec hanics Since the gauge symmetry of electro dynamics w as w ell kno wn, it w as noticed b y the founding fathers that if quan tum mec hanics is to b e coupled to electro dynamics, then the Sc hr odinger equation ( 1.1 ) needs to b e gauge in v arian t as w ell. The most simple w a y of ac hieving this is to require the Hamiltonian app earing in ( 1.1 ) to b e of the form H = [ P q A =c ] 2 2 m + V + q ; (1.8) where P is the quan tum mec hanical momen tum, V is the p oten tial energy and m is the mass of the c harge q : This is in analogy with the Hamiltonian for a classical c harge in the presence of the electromagnetic eld [ 27 28 ]. The coupling sc heme em b o died in ( 1.8 ) is kno wn as minimal c oupling since it is the simplest p ossible gauge in v arian t coupling imaginable. The gauge symmetry inheren t in the com bined system of Sc hr odinger's equation and Maxw ell's equations in p oten tial form can b e generated b y the transformation A A 0 = A + r F 0 = F =c 0 = exp ( iq F =c ) : (1.9) The transformation on the w a v efunction is called a lo c al gauge tr ansformation and diers from the glob al gauge tr ansformation exp ( i ) ; where is a constan t. These

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6 global gauge transformations are irrelev an t in quan tum mec hanics where the w a v efunction is indeterminate up to a global phase. Application of the gauge transformation ( 1.9 ) to the Sc hr odinger equation with Hamiltonian ( 1.8 ) and to Maxw ell's equations in p oten tial form leads to exactly the same equations after the transformation. Therefore, lik e the p oten tial equations ( 1.4 ) b y themselv es, the system of Maxw ell-Sc hr odinger equations i = [ P q A =c ] 2 2 m + V + q (1.10) r 2 A A c 2 r h r A + c i = 4 c J (1.11a) r 2 + r A c = 4 (1.11b) is in v arian t under the gauge transformation ( 1.9 ). There are sev eral other symmetries that are enjo y ed b y this system of equations. F or example, they are in v arian t under spatial rotations, nonrelativistic (Galilei) b o osts, and time rev ersal. As a result, the Maxw ell-Sc hr odinger equations enjo y c harge, momen tum, angular momen tum, and energy conserv ation. That eac h con tin uous symmetry giv es rise to an asso ciated conserv ation la w w as pro v en b y Emm y No ether in 1918 (see Goldstein [ 27 ], Jos e and Saletan [ 28 ], and Abraham and Marsden [ 29 ], and the references therein). This issue is discussed in Chapter 2 in greater detail. It is w orth while men tioning that the Maxw ell-Sc hr odinger equations are obtainable as the nonrelativistic limit of the Maxw ell-Dirac equations i D = mc 2 D + c [ P q A =c ] D + q D (1.12) r 2 A A c 2 r h r A + c i = 4 c J (1.13a) r 2 + r A c = 4 (1.13b)

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7 whic h are the equations of quan tum electro dynamics (QED) [ 19 24 30 ]. Here the w a v efunction D is a 4-comp onen t spinor where the rst t w o comp onen ts represen t the electron and the second t w o comp onen ts represen t the p ositron, eac h with spin1/2. The matrices and are related to the P auli spin matrices [ 7 8 ] and c is the v elo cit y of ligh t. This system of equations p ossesses eac h of the symmetries of the Maxw ell-Sc hr odinger equations and in addition is in v arian t under relativistic b o osts. 1.3 Approac hes to the Solution of the Maxw ell-Sc hr odinger Equations Solving the Maxw ell-Sc hr odinger equations as a coupled and closed system emb o dies the theory of radiation reaction [ 9 26 31 ], whic h is a main theme of this dissertation. Ho w ev er, it should rst b e p oin ted out that ( 1.1 ) and ( 1.2 ) are commonly treated separately In these cases, the eects of one system on the other are handled in one of the follo wing t w o w a ys: The arrangemen t of c harge and curren t is sp ecied and acts as a source for the electromagnetic eld according to ( 1.2 ). The dynamics of the electromagnetic eld is sp ecied and mo dies the dynamics of the matter according to ( 1.1 ). It is not surprising that either of these approac hes is v alid in man y ph ysical situations. Most of the theory of electro dynamics, in whic h the external sources are prescrib ed, ts in to the rst case, while all of classical and quan tum mec hanics in the presence of sp ecied external elds ts in to the second. As a further example of the rst case, the dip ole p o w er radiated b y oscillating dip oles generated b y c harge transfer pro cesses in the in teraction region of p H collisions can b e computed in a straigh tforw ard manner [ 32 33 ]. It is assumed that the dynamics of the oscillating dip ole is kno wn and is used to compute the dip ole radiation, but this radiation do es not inruence the p H collision. As a result energy momen tum, and angular momen tum are not conserv ed b et w een the proton, h ydrogen atom, and electromagnetic eld system. As a further example of the second case, the eects of stim ulated absorption or emission of electromagnetic

PAGE 20

8 radiation b y a molecular target can b e added to the molecular quan tum mec hanics as a rst order p erturbativ e correction. The electro dynamics is sp ecied and p erturbs the molecule but the molecule do es not itself inruence the electro dynamics. This approac h, whic h is kno wn as F ermi's golden rule (see Chapter 3 and Merzbac her [ 7 ], Craig and Thirunamac handran [ 34 ], and Sc hatz and Ratner [ 35 ]) is straigh tforw ard and barring certain restrictions can b e applied to man y ph ysical systems. The system of Maxw ell-Sc hr odinger equations or its relativistic analog can b e closed and is coupled when the Sc hr odinger w a v efunction ; whic h is the solution of ( 1.1 ), is c hosen to b e the source for the scalar p oten tial and v ector p oten tial A in ( 1.4 ). In particular, the sources of c harge and curren t J ; whic h pro duce the electromagnetic p oten tials according to ( 1.4 ), in v olv e the solutions of the Sc hr odinger equation according to = q J = q [ i r q A =c ] + [ i r q A =c ] = 2 m: (1.14) On the other hand, the w a v efunction is inruenced b y the p oten tials that app ear in the Hamiltonian H in ( 1.8 ). The in terpretation of the Sc hr odinger w a v efunction as the source for the electromagnetic eld w as Schr odinger's ele ctr omagnetic hyp othesis whic h dates bac k to 1926. The disco v ery of the quan tum mec hanical con tin uit y equation and its similarit y to the classical con tin uit y equation of electro dynamics only reinforced the h yp othesis. Ho w ev er, it implied the electron to b e smeared out throughout the atom and not lo cated at a discrete p oin t, whic h is in con tradiction to the accepted Born probabilistic or Cop enhagen in terpretation. Sc hr odinger's w a v e mec hanics had some success, esp ecially with the in teraction of the electromagnetic eld with b ound states, but failed to prop erly describ e scattering states due to the probabilistic nature of measuremen t of the w a v efunction. In addition, certain prop erties of electromagnetic radiation w ere found to b e inconsisten t with exp erimen t.

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9 Sc hr odinger's electromagnetic h yp othesis w as extended b y F ermi in 1927 and later b y Crisp and Ja ynes in 1969 [ 36 ] to incorp orate the unquan tized electromagnetic self-elds in to the theory That is, the classical electromagnetic elds produced b y the atom w ere allo w ed to act bac k up on the atom. The solutions of this extended semiclassical theory captured certain asp ects of sp on taneous emission as w ell as frequency shifts lik e the Lam b shift. Ho w ev er, it w as quic kly noticed that some deviations from QED existed [ 37 ]. F or example, F ermi's and Ja ynes's theories predicted a time-dep enden t form for sp on taneous deca y that is not exp onen tial. There are man y prop erties that are correctly predicted b y this semiclassical theory and are also in agreemen t with QED. In the cases where the semiclassical theory disagrees with QED [ 37 ], it has alw a ys b een exp erimen tally v eried that QED is correct. Nev ertheless, the semiclassical theory do es not suer from the mathematical and logical diculties that are presen t in QED. T o this end, the semiclassical theory when it is correct, pro vides a useful alternativ e to the quan tum eld theory It is generally simpler and its solutions pro vide a more detailed dynamical description of the in teraction of an atom with the electromagnetic eld. Since 1969 man y others ha v e follo w ed along the semiclassical path of Crisp and Ja ynes. Nesb et [ 38 ] computed the gauge in v arian t energy pro duction rate from a man y particle system. Co ok [ 39 ] used a densit y op erator approac h to accoun t for sp on taneous emission without lea ving the atomic Hilb ert space. Barut and V an Huele [ 40 ] and Barut and Do wling [ 41 42 ] form ulated a self-eld quan tum electrodynamics for Sc hr odinger, P auli, Klein-Gordon, and Dirac matter theories. They w ere able to eliminate all electromagnetic v ariables in fa v or of Green's function in tegrals o v er the sources and w ere able to reco v er the correct exp onen tial sp on taneous deca y from an excited state. Some p ertinen t critiques of this w ork are expressed b y Bialynic ki-Birula in [ 43 ] and b y Crisp in [ 44 ]. Bosanac [ 45 { 47 ] and Do sli c and

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10 Bosanac [ 48 ] argued that the instan taneous eects of the self in teraction are unph ysical. As a result, they form ulated a theory of radiation reaction based on the retarded eects of the self-elds. Milonni, Ac k erhalt, and Galbraith [ 49 ] predicted c haotic dynamics in a collection of t w o-lev el atoms in teracting with a single mo de of the classical electromagnetic eld. Crisp himself has con tributed some of the nest w ork in semiclassical theory He computed the radiation reaction asso ciated with a rotating c harge distribution [ 50 ], the atomic radiativ e lev el shifts resulting from the solution of the semiclassical nonlinear in tegro-dieren tial equations [ 51 ], the in teraction of an atomic system with a single mo de of the quan tized electromagnetic eld [ 52 53 ], and the extension of the semiclassical theory to include relativistic eects [ 54 ]. Besides semiclassical theory a v ast amoun t of researc h has b een conducted in the quan tum theory of electro dynamics and matter. QED [ 19 24 30 55 ] (see Chapter 3 ), whic h is the fully relativistic and quan tum mec hanical theory of electrons and photons, has b een found to agree with all asso ciated exp erimen ts. The coupled equations of QED can b e solv ed nonp erturbativ ely [ 56 57 ], but are most often solv ed b y resorting to p erturbativ e metho ds. As w as previously men tioned, there are some dra wbac ks to these metho ds that are not presen t in the semiclassical theory In addition to pure QED in terms of electrons and photons, there has also b een an increasing in terest in molecular quan tum electro dynamics [ 34 ]. P o w er and Thirunamac handran [ 58 59 ], Salam and Thirunamac handran [ 60 ], and Salam [ 61 ] ha v e used p erturbativ e metho ds within the minimal-coupling and m ultip olar formalisms to study the quan tized electromagnetic eld surrounding a molecule. In particular, they ha v e claried the relationship b et w een the t w o formalisms and in addition ha v e calculated the P o yn ting v ector and sp on taneous emission rates for magnetic dip ole and electric quadrup ole transitions in optically activ e molecules.

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11 In b oth the semiclassical and quan tum mec hanical con text the self-energy of the electron has b een studied [ 62 { 65 ]. The self-energy arises naturally in the minimal coupling sc heme as the q term in the Hamiltonian ( 1.8 ). More sp ecically the electron's self-energy in the nonrelativistic theory is dened as U = R V d 3 xq ( x ; t ) ( x ; t )( x ; t ) = R V d 3 x R V d 3 x 0 ( x ; t ) ( x 0 ; t ) j x x 0 j : (1.15) As a result of the q term, the Sc hr odinger equation ( 1.10 ) is nonlinear in : It resem bles the nonlinear Sc hr odinger equation [ 66 ] i u = a ( d 2 u=dx 2 ) + b j u j 2 u (1.16) whic h arises in the mo deling of Bose-Einstein condensates with the Gross-Pitaevskii equation and in the mo deling of sup erconductivit y with the Ginzburg-Landau equation. In the relativistic theory the electron is forced to ha v e no structure due to relativistic in v ariance. As a result, the corresp onding self-energy is innite. On the other hand, the electron ma y ha v e structure in the nonrelativistic theory Consequen tly the self-energy is nite. The self-energy of the electron will b e discussed in Chapter 4 in more detail. 1.4 Canonical F orm ulation of the Maxw ell-Sc hr odinger Equations The w ork presen ted in this dissertation [ 67 ] con tin ues the semiclassical story originally form ulated b y F ermi, Crisp, and Ja ynes. Unlik e other semiclassical and quan tum mec hanical theories of electro dynamics and matter where the gauge is xed at the b eginning, it will b e emphasized that the gauge is a fundamen tal degree of freedom in the theory and should not b e eliminated. As a result, the equations of motion are naturally w ell-balanced and form a w ell-dened initial v alue problem when the gauge freedom is retained. This philosoph y w as pursued early on b y Dirac, F o c k, and P o dolsky [ 68 ] (see Sc h winger [ 19 ]) in the con text of the Hamiltonian

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12 form ulation of QED. Ho w ev er, their approac h w as quic kly forgotten in fa v or of the more practical Lagrangian based p erturbation theory that no w dominates the QED comm unit y More recen tly Kob e [ 69 ] studied the Hamiltonian approac h in semiclassical theory Unfortunately he did not recognize the dynamical equation asso ciated with the gauge and refers to it as a meaningless equation. It is b eliev ed that the Hamiltonian form ulation of dynamics oers a natural and p o w erful theoretical approac h to the in teraction of electro dynamics and w a v e mec hanics that has not y et b een fully explored. T o this end, the Hamiltonian or canonical form ulation of the Maxw ell-Sc hr odinger dynamics is constructed in this dissertation. ( Canonic al means according to the canons, i.e. standard or con v en tional.) The asso ciated w ork in v olv es nonp erturbativ e analytic and n umerical metho ds for the solution of the coupled and closed nonlinear system of Maxw ell-Sc hr odinger equations. The rexibilit y inheren t in these metho ds captures the nonlinear and nonadiabatic eects of the coupled system and has the p oten tial to describ e situations where the atomic and electromagnetic dynamics o ccur on the same timescale. The canonical form ulation is set up b y applying the time-dep enden t v ariational principle to the Sc hr odinger Lagrangian L Sc h = i [ i r q A =c ] [ i r q A =c ] 2 m V q ; (1.17) and Maxw ell Lagrangian together with a Lorenz gauge xing term, i.e. L LMax = L Max [ =c + r A ] 2 8 = [ A =c r ] 2 [ r A ] 2 8 [ =c + r A ] 2 8 : (1.18) This yields a set of coupled nonlinear rst order dieren tial equations of the form = @ H =@ (1.19)

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13 where is a symplectic form, is a column v ector of the dynamical v ariables, and H is the Maxw ell-Sc hr odinger Hamiltonian (see Chapter 4 ). These matrix equations form a w ell-dened initial v alue problem. That is, the solution to these equations is uniquely determined and kno wn in principle once the initial v alues for eac h of the dynamical v ariables are sp ecied. These equations are also closed since the Sc hr odinger w a v efunction acts as the source, whic h is nonlinear (see J in 1.14 ), for the electromagnetic p oten tials and these p oten tials act bac k up on the w a v efunction. By represen ting eac h of the dynamical v ariables in a basis of gaussian functions G K i.e. ( x ; t ) = P K G K ( x ) K ( t ) ; where the time-dep enden t sup erp osition co ecien ts K ( t ) carry the dynamics, the time-dep enden t v ariational principle generates a hierarc h y of appro ximations to the coupled Maxw ell-Sc hr odinger equations. In the limit of a complete basis these equations reco v er the exact Maxw ell-Sc hr odinger theory while in an y nite basis they form a basis represen tation that can systematically b e made more accurate with a more robust basis. The asso ciated basis equations ha v e b een implemen ted in a F or tran 90 computer program [ 70 ] that is rexible enough to handle arbitrarily man y gaussian basis functions, eac h with adjustable widths and cen ters. In addition, a no v el n umerical con v ergence accelerator has b een dev elop ed based on remo ving the large Coulom bic elds surrounding a c harge (that can b e computed analytically from Gauss's la w, i.e. r E = r 2 = 4 ; once the initial conditions are pro vided) b y applying a certain canonical transformation to the dynamical equations. The canonical transformation separates the dynamical radiation from the Coulom bic p ortion of the eld. This in turn allo ws the basis to describ e only the dynamics of the radiation elds and not the large Coulom bic eects. The canonical transformed equations, whic h are of the form ~ ~ = @ ~ H =@ ~ ; ha v e b een added to the existing computer program and the con v ergence of the solution of the Maxw ell-Sc hr odinger equations is studied.

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14 The canonical approac h to dynamics enjo ys a deep mathematical foundation and p ermits a general application of the theory to man y ph ysical problems. In particular, the dynamics of the h ydrogen atom in teracting with its electromagnetic eld has b een in v estigated for b oth stationary and sup erp ositions of stationary states. Stationary state solutions of the com bined h ydrogen atom and electromagnetic eld system as w ell as nonstationary states that pro duce electromagnetic radiation ha v e b een constructed. This radiation carries a w a y energy momen tum, and angular momen tum from the h ydrogen atom suc h that the total energy momen tum, angular momen tum, and c harge of the com bined system are conserv ed. A series of plots are presen ted to highligh t this atom-eld dynamics. 1.5 F ormat of Dissertation A tour of the Lagrangian and Hamiltonian dynamics is presen ted in Chapter 2 Hamilton's principle is applied to the deriv ation of the Euler-Lagrange equations of motion. Emphasis is placed on the Hamiltonian form ulation of dynamics, whic h is presen ted from the mo dern p oin t of view whic h mak es connection with symplectic geometry T o this end, b oth conguration space and phase space geometries are discussed. In Chapter 3 the Sc hr odinger and Maxw ell dynamics will b e presen ted from the p oin t of view of p erturbation theory In the Sc hr odinger theory the electromagnetic eld is treated as a p erturbation on the stationary states of an atomic or molecular system. In the long time limit, the F ermi golden rule accoun ts for stim ulated transitions b et w een these states. As an example, the absorption cross section is calculated for an atom in the presence of an external eld. QED is discussed to emphasize the success of p erturbation theory In the Maxw ell theory the electromagnetic elds arising from sp ecied sources of c harge and curren t are presen ted. The rst order (electric dip ole) m ultip olar con tributions to the electromagnetic eld are calculated. Lastly the bremsstrahlung from a gaussian c harge distribution is analyzed.

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15 Chapter 4 con tains the main b o dy of the dissertation, whic h is on the Hamiltonian or canonical approac h to the Maxw ell-Sc hr odinger dynamics. Nonp erturbativ e analytic metho ds are constructed for the solution of the asso ciated coupled and nonlinear equations. The gauge symmetry is discussed in detail and exploited to cast the Maxw ell-Sc hr odinger equations in to a w ell-dened initial v alue problem. The theory of canonical or symplectic transformations is used to construct a sp ecial transformation to remo v e the Coulom bic con tribution to the dynamical v ariables. The w ell-dened Maxw ell-Sc hr odinger theory from Chapter 4 is n umerically implemen ted in Chapter 5 The asso ciated equations of motion are expanded in to a basis of gaussian functions, whic h renders the partial dieren tial equations as ordinary dieren tial equations. These equations are co ded in F or tran 90. In addition, the (canonical transformed) equations asso ciated with the Coulom b reference are incorp orated in to the existing co de. The dynamics of the spinless h ydrogen atom in teracting with the electromagnetic eld are presen ted in a series of plots. A summary and conclusion of the dissertation are presen ted in Chapter 6 1.6 Notation and Units A brief statemen t should b e made ab out notation. All w ork will b e done in the (1+3)-dimensional bac kground of sp ecial relativit y with diagonal metric tensor g = g with elemen ts g 00 = g 00 = 1 and g 11 = g 22 = g 33 = 1 : All 3-v ectors will b e written in b old faced Roman while all 4-v ectors will b e written in italics. As usual, Greek indices run o v er 0, 1, 2, 3 or ct; x; y ; z and Roman indices run o v er 1, 2, 3 or x; y ; z The summation con v en tion is emplo y ed o v er rep eated indices. F or example, the 4-p oten tial A = ( A 0 ; A k ) = ( ; A ) and A = g A = ( ; A ) : The D'Alem b ertian op erator = r 2 @ 2 =@ ( ct ) 2 = @ 2 is used at times in fa v or of @ 2 : F ourier transforms will b e denoted with tildes, e.g. ~ F is the F ourier transform of F : The represen tation indep enden t Dirac notation j h i will b e used in the discussion of time-dep enden t p erturbation theory but for the most part functions h ( x ) = h x j h i

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16 or ~ h ( k ) = h k j h i will b e used. (It will b e assumed that all of the functions of ph ysics are in C 1 and in L 2 \ L 1 o v er either the real or complex eld.) Since it is the radiation eects presen t on the atomic scale that are of in terest, it is b enecial to w ork in natural (gaussian atomic) units where ~ = j e j = m e = 1 : In these units the sp eed of ligh t c 137 atomic units of v elo cit y

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CHAPTER 2 THE D YNAMICS A dynamical system ma y b e w ell-dened once its Lagrangian and asso ciated dynamical v ariables as w ell as their initial v alues are sp ecied. This information together with the calculus of v ariations [ 71 ] generates the equations whic h go v ern the dynamics. Chapter 2 will detail the asp ects asso ciated with generating equations of motion for dynamical systems. Man y dieren t v ariational metho ds exist b y whic h to generate dynamical equations, eac h ha ving subtle dierences [ 72 ]. Ho w ev er, all metho ds rely on the mac hinery inheren t in v ariational calculus. Giv en a starting and ending p oin t for the dynamics, the calculus of v ariations determines the path connecting them. The dynamics is determined b y extremizing (either minimizing or maximizing) a certain function of these initial and nal p oin ts. In this c hapter, the Lagrangian and Hamiltonian formalisms [ 27 { 29 ] are presen ted for discrete and con tin uous systems. The Lagrangian approac h leads to second order equations of motion in time, while the Hamiltonian or canonical approac h leads to rst order equations of motion in time. The resulting dynamics are equiv alen t in either case. Ho w ev er, the Hamiltonian approac h enjo ys a ric h mathematical foundation connecting dieren tial geometry and dynamics [ 28 29 ]. Muc h of the remainder of this dissertation will b e dev oted to the canonical form ulation of Maxw ell and Sc hr odinger theories. The time-dep enden t v ariational principle [ 73 ], whic h has its origin in n uclear ph ysics [ 74 ], is the v ariational approac h to the determination of the Sc hr odinger equation. The Hamiltonian dynamics asso ciated with the Sc hr odinger equation ev olv es in a generalized phase space endo w ed with a P oisson brac k et. With the 17

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18 time-dep enden t v ariational principle, man y-b o dy dynamics ma y b e consisten tly describ ed in terms of a few ecien tly c hosen dynamical v ariables (see Deumens et. al. [ 75 ]). Additionally the v ariational tec hnology pro vides a means b y whic h to construct appro ximations to the resulting equations of motion in a systematic and w ell-balanced w a y As will b e seen in Chapters 4 and 5 these appro ximations will b e of utmost imp ortance in the n umerical solutions of the coupled nonlinear Maxw ellSc hr odinger equations. 2.1 Lagrangian F ormalism Before delving in to a detailed accoun t of Lagrangian dynamics it is instructiv e to sa y a few w ords ab out the Lagrangian itself. The Lagrangian is a scalar function of the v ectors q k and q k ( k = 1 ; : : : ; N ) with dimensions of energy Ho w ev er, it is not the energy nor is it ph ysically observ able. The Lagrangian is a fundamen tal ingredien t in the determination of a dynamical system. That is, the dynamics of a system ma y b e kno wn in principle once the system's Lagrangian is kno wn and the dynamical v ariables are giv en at some time. The Lagrangian ma y ha v e a n um b er of symmetries. In 1918, Emm y No ether (see Goldstein [ 27 ] and the references therein) pro v ed that to eac h con tin uous symmetry there is an asso ciated conserv ation la w. F or example, since all observ ations indicate that Nature is in v arian t under time and space translations as w ell as spatial rotations, so should b e the Lagrangian. If the Lagrangian p ossesses time translation in v ariance, then the energy of the system is conserv ed. If the Lagrangian is in v arian t to space translations (rotations), then the linear (angular) momen tum of the system is conserv ed. One last symmetry of signicance in this dissertation is the gauge symmetry Since Nature is in v arian t to the c hoice of gauge, the Lagrangian should main tain this symmetry as w ell. If the gauge symmetry is preserv ed, then the system enjo ys conserv ation of c harge. Dep ending on the particular system at

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19 hand, other symmetries ma y b e of imp ortance and should also b e resp ected b y the Lagrangian.2.1.1 Hamilton's Principle Giv en a Lagrangian L ( q k ; q k ; t ) dep enden t up on the N p osition v ectors q k ; the N v elo cit y v ectors q k ; and also the time t; the action I is dened b y the path in tegral I ( q k ; q k ; t ) R t 2 t 1 L ( q k ; q k ; t ) dt k = 1 ; : : : ; N : (2.1) That the v ariation of this in tegral b et w een the xed times t 1 and t 2 leads to a stationary p oin t is a statemen t of Hamilton 's Principle [ 27 28 ]. Moreo v er, this stationary p oin t is the correct path for the motion. In mathematical sym b ols, the motion is a solution of I = R t 2 t 1 Ldt = 0 ; (2.2) where I is the variation of the action I : Only those paths are v aried for whic h q k ( t 1 ) = 0 = q k ( t 2 ) : A particular form of the v ariational path parametrized b y the innitesimal parameter is giv en b y q k ( t; ) = q k ( t; 0) + k ( t ) ; (2.3) where q k ( t ) = q k ( t; 0) is the correct path of the motion and the v ectors k ( t ) are w ell-b eha v ed and v anish at the b oundaries t 1 and t 2 : By con tin uously deforming q k ( t; ) un til it is extremized, the correct path can b e found. This parametrization of the path in turn parametrizes the action itself. Equation ( 2.2 ) ma y no w b e rewritten more precisely as I ( ) = @ I ( ) @ =0 d = 0 (2.4)

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20 whic h represen ts innitesimal v ariations from the correct path. The calculus of v ariations yields @ I ( ) @ = R t 2 t 1 dt n @ L @ q k @ q k @ + @ L @ q k @ q k @ o = @ L @ q k @ q k @ t 2 t 1 + R t 2 t 1 dt n @ L @ q k d dt @ L @ q k o @ q k @ ; (2.5) where a partial in tegration w as p erformed in the second line. Since q k ( t 1 ) = 0 = q k ( t 2 ) ; the surface term v anishes. The stationary p oin t of the v ariation is therefore determined b y R t 2 t 1 dt n @ L @ q k d dt @ L @ q k o @ q k @ =0 = 0 : (2.6) But since the v ectors @ q k =@ are arbitrary (c ho ose in particular @ q k =@ > 0 and con tin uous on [ t 1 ; t 2 ]), the in tegral is zero only when @ L @ q k d dt @ L @ q k = 0 (2.7) b y the fundamen tal lemma of the calculus of v ariations. Equation ( 2.7 ) denes the system of N second order Euler-L agr ange dier ential e quations in terms of the lo cal co ordinates ( q k ; q k ) : Since these equations are v alid on ev ery co ordinate c hart, the Euler-Lagrange equations are co ordinate indep enden t. It is demonstrated in [ 28 ] that ( 2.7 ) can b e written in a co ordinate free or purely geometric form. If these equations admit a solution, then the action has a stationary v alue. It is this stationary v alue whic h determines the motion. The second order form of the EulerLagrange equations can b e seen b e expanding the total time deriv ativ e to giv e @ L @ q k n @ 2 L @ q l @ q k q l + @ 2 L @ q l @ q k q l + @ 2 L @ t@ q k o = 0 : (2.8) It will alw a ys b e assumed unless otherwise noted that the Hessian condition is satised. That is det f @ 2 L =@ q l @ q k g 6 = 0 : Lastly notice that the Lagrangian is arbitrary up to the addition of a total time deriv ativ e. That is, if L L 0 = L + ( d=dt ) K for K a w ell-b eha v ed function of the

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21 dynamical v ariables, then the action I R t 2 t 1 f L + ( d=dt ) K g dt = K ( t 2 ) K ( t 1 ) + R t 2 t 1 L dt = R t 2 t 1 L dt = I (2.9) since K ( t 2 ) = 0 = K ( t 1 ) : Th us the same Euler-Lagrange equations ( 2.7 ) are generated for L 0 as for L: In other w ords, there are man y Lagrangians that lead to the same equations of motion. There is no unique Lagrangian for a particular dynamical system. All Lagrangians diering b y only a time deriv ativ e will lead to the same dynamics. More generally in the dynamics of con tin uous systems t w o equiv alen t Lagrangians ma y dier b y a purely surface term in time and space. 2.1.2 Example: The Harmonic Oscillator in ( q k ; q k ) It is no w useful to presen t a brief illustrativ e example. In t w o freedoms, the dynamics of a scalar mass sub jected to the force of a harmonic p oten tial with frequency k is determined b y the Lagrangian (no summation) L ( q k ; q k ) = 1 2 m q k q k 1 2 m! 2 k q k q k k = 1 ; 2 (2.10) whic h is a function of the real-v alued v ectors q k and q k : Application of the calculus of v ariations to the asso ciated action functional leads to ( 2.7 ) with @ L=@ q k = m q k and @ L=@ q k = m! 2 k q k : The second order Euler-Lagrange equations of motion are 1 2 m q k + 1 2 m! 2 k q k = 0 k = 1 ; 2 (2.11) with initial v alue solution q k ( t ) = q k ( t 0 ) cos( k t ) + q k ( t 0 ) sin ( k t ) =! k : It is said that q k is an in tegral curv e of the dynamical equation ( 2.11 ). Once the initial v alues q k ( t 0 ) and q k ( t 0 ) are pro vided, the dynamics of the harmonic oscillator is kno wn. This dynamics o ccurs in a space whose co ordinates are not just the q k ; but b oth the q k and q k : Some geometric asp ects of this space will no w b e presen ted.

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22 PSfrag replacemen ts Q T q k Q q k Figure 2{1: The conguration manifold Q = S 2 is depicted together with the tangen t plane T q k Q at the p oin t q k 2 Q : 2.1.3 Geometry of T Q In the Lagrangian formalism, the dynamics unfolds in a v elo cit y phase space whose p oin ts are of the form ( q k ; q k ) : The p osition v ectors q k lie in a dieren tial manifold called the conguration manifold Q ; while the v elo cit y v ectors lie in the manifold of v ectors tangen t to Q : The space formed b y attac hing the space spanned b y all v ectors tangen t to the p oin t q k 2 Q is called the tangent b er ab ove q k or the tangent plane at q k and is denoted b y T q k Q : The union of the conguration manifold Q and the collection of all b ers T q k Q for eac h p oin t q k 2 Q (together with lo cal c harts on T q k Q ) is called the velo city phase sp ac e tangent bund le or tangent manifold of Q and is denoted b y T Q : It is that manifold that carries the Lagrangian dynamics, not the conguration manifold Q : A picture is presen ted in Figure 2{1 corresp onding to the case where Q is the t w o-dimensional surface S 2 of the unit ball in R 3 : The tangen t plane at the p oin t q k reac hes out of S 2 and in to R 3 : This larger manifold is where the asso ciated Lagrangian dynamics o ccurs.

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23 The in tegral curv es of a dynamical system are v ector elds and are called the dynamics or the dynamic al ve ctor elds The v elo cit y phase space dynamics is a ve ctor eld on T Q denoted b y L q k ( @ =@ q k ) + q k ( @ =@ q k ) ; where q k and q k are the comp onen ts of L and @ =@ q k and @ =@ q k form a lo cal basis for L : The time dep endence of a dynamical v ariable F ( q k ; q k ) ; whic h is an implicitly time-dep enden t function on T Q ; is determined b y its v ariation along the dynamics. That is F ( q k ; q k ) L ( F ) = @ F @ q k q k + @ F @ q k q k : (2.12) The accelerations q k can b e substituted directly from the dynamical equations. Th us, the time dep endence of a dynamical v ariable is determined b y the equations of motion themselv es without ev en the kno wledge of their solution. Bey ond functions and v ector elds on T Q ; there is another imp ortan t geometrical quan tit y called the one-form that is w orth dening. One-forms on T Q are linear functionals that map v ector elds to functions. That is, if the one-form = A 1a dq a + A 2a d q a is applied to the v ector eld X = X b 1 ( @ =@ q b ) + X b 2 ( @ =@ q b ) ; then their inner pro duct results in h j X i = A 1a X b 1 dq a ( @ =@ q b ) + A 1a X b 2 dq a ( @ =@ q b ) + A 2a X b 1 d q a ( @ =@ q b ) + A 2a X b 2 d q a ( @ =@ q b ) = A 1a X b 1 a b + A 2a X b 2 a b = A 1a X a 1 + A 2a X a 2 ; (2.13) where dq a ( @ =@ q b ) = d q a ( @ =@ q b ) = a b and dq a ( @ =@ q b ) = d q a ( @ =@ q b ) = 0 ; and where A a and X b are the lo cal comp onen ts, whic h are functions, of the one-form and v ector eld X : It is common to write h j X i ( X ) : It should b e p oin ted out that the dieren tial of a function is a one-form. That is dF = @ F @ q k dq k + @ F @ q k d q k (2.14)

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24 is a one-form and ma y b e applied to the dynamical v ector eld L to giv e dF ( L ) h dF j L i = @ F @ q k q k + @ F @ q k q k = F : (2.15) The one-forms are also called c ovariant ve ctors or c ove ctors and are dual to the v ector elds whic h are sometimes called c ontr avariant ve ctors. 2.2 Hamiltonian F ormalism The Lagrangian formalism set up N second order dynamical equations whic h required 2 N initial v alues to x the dynamics. Alternativ ely and equiv alen tly the dynamics ma y b e describ ed in terms of 2 N rst order equations of motion with 2 N initial v alues. This so called Hamiltonian dynamics ev olv es in a dieren t tangen t manifold or phase space with generalized co ordinates q a and p a ; whic h are go v erned b y the dynamical equations q a = @ H @ p a and p a = @ H @ q a ; (2.16) where the function H is called the Hamiltonian (see ( 2.18 ) b elo w). It is itself a dynamical v ariable and for man y ph ysical systems it is the energy Since ( 2.16 ) are of rst order, the asso ciated tra jectories are separated on the new phase space. The c hange of v ariables from ( q a ; q a ) to ( q a ; p a ) is accomplished b y a Legendre transformation [ 27 28 ]. The momen tum conjugate to the v ector q a is dened in terms of the Lagrangian L b y p a @ L ( q a ; q a ) @ q a : (2.17) Notice that this c onjugate momentum is not a v ector as is the v elo cit y q a and do es not lie in the tangen t manifold T Q : Rather the momen tum p a is dual to the p osition v ector q a : It is a one-form and lies in the cotangen t manifold T Q : This dierence will so on b e elab orated on. With the momen tum p a and the Lagrangian, the Hamiltonian

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25 function is constructed according to H ( q a ; p a ) = p a q a ( q a ; p a ) L ( q a ; p a ) : (2.18) Here it is assumed that the relation ( 2.17 ) can b e in v erted to solv e for the v elo cit y q a : Hamilton's canonical equations of motion ( 2.16 ), whic h are rst order dieren tial equations in time, can no w b e obtained from an argumen t similar to that presen ted in Section 2.1.1 on Hamilton's Principle. That is, if the Lagrangian in the action in tegral ( 2.1 ) is replaced b y L = p a q a H from ( 2.18 ), i.e. I ( q a ; p a ) R t 2 t 1 [ p a q a ( q a ; p a ) H ( q a ; p a )] dt; (2.19) then Hamilton's equations follo w in a straigh tforw ard manner. 2.2.1 Example: The Harmonic Oscillator in ( q a ; p a ) It is no w useful to compare the Lagrangian and Hamiltonian dynamics for a simple dynamical system. Recall the Lagrangian for the t w o freedom harmonic oscillator in ( 2.10 ). That is L ( q a ; q a ) = 1 2 m q a q a 1 2 m! 2 a q a q a a = 1 ; 2 : ( 2.10 ) The momen tum conjugate to q a is p a @ L =@ q a = m q a and with it the Hamiltonian b ecomes H ( q a ; p a ) = p a q a n p a p a 2 m 1 2 m! 2 a q a q a o = p a p a 2 m + m! 2 a q a q a 2 : (2.20) With this Hamiltonian the equations of motion are: q a = @ H @ p a = p a =m p a = @ H @ q a = m! 2 a q a (2.21) and ha v e the initial v alue solutions q a ( t ) = q a ( t 0 ) cos( a t ) + q a ( t 0 ) sin( a t ) =! a and p a ( t ) = p a ( t 0 ) cos( a t ) m! a q a ( t 0 ) sin ( a t ) : These are the in tegral curv es of the rst

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26 order dieren tial equations ( 2.21 ) and ma y b e compared to those in the Lagrangian form ulation. 2.2.2 Symplectic Structure and P oisson Brac k ets One of the man y p o w erful asp ects of the Hamiltonian or canonical approac h to dynamics is the rexibilit y and abilit y to treat p ositions and momen ta similarly This similarit y among the co ordinates is made explicit b y the follo wing notation: a = q a a = 1 ; : : : ; N a = p a N a = N + 1 ; : : : ; 2 N : (2.22) Similarly the forces b ecome @ H =@ p a @ H =@ a + N and @ H =@ q a @ H =@ a so that the equations of motion are: a = @ H @ a + N a = 1 ; : : : ; N a = @ H @ a N a = N + 1 ; : : : ; 2 N : (2.23) These Hamilton equations ma y b e written more compactly as ab b = @ H @ a ; (2.24) where ab are the matrix elemen ts of the symple ctic form : The symplectic form is an an tisymmetric 2 N 2 N -dimensional matrix of the form = 0 N 1 N 1 N 0 N ; (2.25) where 0 N and 1 N are the N N -dimensional zero and iden tit y matrices resp ectiv ely The matrix ( 2.25 ) is also referred to as the c anonic al symple ctic form b ecause it satises the prop erties: 2 = 1 and T = ; (2.26) or equiv alen tly ab bc = a c and ab = ba : The matrix elemen t ab with b oth indices up is the in v erse of ab :

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27 In ( 2.12 ), the time deriv ativ e or v ariation of a (implicitly time-dep enden t) dynamical v ariable F on T Q w as demonstrated. In a similar fashion, F can b e view ed in the momen tum phase space T Q ; whic h will b e discussed shortly It is F ( q b ; p b ) = @ F @ b b = @ F @ b ba @ H @ a ; (2.27) where the equation of motion ( 2.24 ) w as in v erted and substituted for b : The righ t hand side of this equation is called the Poisson br acket of F with H : In general, it ma y b e written for an y t w o functions in T Q as f F ; G g @ F @ b ba @ G @ a = @ F @ q a @ G @ p a @ F @ p a @ G @ q a : (2.28) In particular, an alternativ e form of Hamilton's equations is deriv ed when the P oisson brac k et is applied to the co ordinate : That is a = f a ; H g : (2.29) Since the P oisson brac k et is bilinear, an tisymmetric, and satises the Jacobi iden tit y f f ; g h g = g f f ; h g + f f ; g g h; the set of functions on T Q forms a Lie algebra under P oisson brac k et f ; g : In fact, the Hamiltonian dynamics can naturally b e studied from this p oin t of view [ 29 73 ]. 2.2.3 Geometry of T Q As w as previously men tioned, the dynamics asso ciated with Hamilton's equations of motion ( 2.24 ) do not unfold in the same v elo cit y phase space T Q that w as dened in Section 2.1.3 These equations of motion dene a v ector eld on a differen t phase space whose comp onen ts are the functions ba ( @ H =@ a ) : The in tegral curv es of this v ector eld are the dynamics. Recall that the p oin ts of T Q are made up of q k and q k : The v elo cities q k are the lo cal comp onen ts of the v ector eld q k ( @ =@ q k ) : Ho w ev er, the momen ta are the lo cal comp onen ts of the one-form p a dq a ( @ L=@ q a ) dq a ; whic h are not the comp onen ts

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28 of a v ector eld. Since one-forms are dual to v ector elds, p a dq a lies in the dual space of T q a Q : This space is the c otangent sp ac e at q a and is denoted b y T q a Q : In analogy with T Q ; the c otangent bund le or c otangent manifold T Q is made up of Q together with its cotangen t spaces T q a Q : Consequen tly the carrier manifold for the Hamiltonian dynamics is not T Q ; but rather it is the phase sp ac e T Q : The dynamical v ector eld on T Q is giv en b y H b @ @ b = q a @ @ q a + p a @ @ p a = @ H @ p a @ @ q a @ H @ q a @ @ p a ; (2.30) where Hamilton's equations of motion ( 2.16 ) w ere substituted for the q a and p a : There is one last geometric quan tit y that needs to b e dened. The symplectic form is a t w o-form on T Q : Two-forms are bilinear, an tisymmetric forms that map pairs of v ector elds to functions. That is, if X = X a ( @ =@ a ) and Y = Y b ( @ =@ b ) are v ector elds on T Q ; then ( X ; Y ) = X a Y b ( @ =@ a ; @ =@ b ) = X a ab Y b = X a Y a Y a X a : (2.31) The matrix elemen ts ab = ba are iden tical to those presen ted earlier. Since is nonsingular and the dieren tial d! = 0 ; i.e. is closed, the t w o-form is called a symple ctic form. In general, phase space is naturally endo w ed with a symplectic form or structure. F or this reason T Q is also a symple ctic manifold [ 29 ]. Lastly it should b e men tioned that ( X ; Y ) is a measure of the area b et w een the v ectors X and Y : In fact, there is a p o w erful theorem attributed to Liouville [ 27 { 29 ] that states that the phase space v olume m ust b e in v arian t under canonical transformations in phase space. Canonical transformations are those transformations that main tain the symplectic structure of Hamilton's dynamical equations ab b = @ H @ a : ( 2.24 ) More will b e said on canonical transformations in Chapter 4

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CHAPTER 3 ELECTR OD YNAMICS AND QUANTUM MECHANICS The coupling of electro dynamics to c harged matter is a complicated problem. This complexit y is comp ounded b y the fact that the elds pro duced b y c harges in motion react bac k up on the c harges, th us causing a mo dication of their tra jectory As men tioned in the in tro duction, the corresp onding ph ysics is often analyzed in one of t w o w a ys. Either: The electromagnetic eld is tak en as an inruence on the dynamics of the c harges. The sources of c harge and curren t are used to calculate the dynamics of the electromagnetic eld. Chapter 3 will discuss b oth of these cases in detail. The rst p ortion of this c hapter will set up the time-dep enden t p erturbation theory whic h will b e used to mak e calculations in quan tum mec hanics under the inruence of an electromagnetic eld. The second p ortion of this c hapter will explore the electro dynamics resulting from a giv en and J : In particular, the m ultip ole expansion will b e in tro duced and used to calculate the p o w er radiated from an oscillating electric dip ole. Additionally the electromagnetic elds corresp onding to a gaussian w a v epac k et will b e presen ted. In the narro w width limit of the gaussian, the resulting ph ysics reduces to the exp ected textb o ok results for a p oin t source. 3.1 Quan tum Mec hanics in the Presence of an Electromagnetic Field The dynamics of c harges in an external electromagnetic eld ma y b e studied at v arying lev els of sophistication from a purely classical description of b oth c harge and eld to a fully quan tum treatmen t. V arious semiclassic al or mixed quan tummatter/classical-eld descriptions are a v ailable as w ell as fully quan tum and relativistically in v arian t treatmen ts suc h as quan tum electro dynamics. 29

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30 Time-dep enden t p erturbation theory [ 7 ] is a systematic metho d b y whic h to calculate (among other things) prop erties of the dynamics of c harges in an external electromagnetic eld. In this section, the time-dep enden t p erturbation theory is intro duced for a general p erturbation in the con text of quan tum mec hanics. Emphasis is then placed on the classical electromagnetic eld as a particular time-dep enden t p erturbation V : Within this framew ork the p erturbation is seen as causing transitions b et w een t w o stationary states j k i and j m i of an atomic system, and is sym b olized to lo w est order b y the matrix elemen t V k m = h k j V j m i : Exp erimental observ ables suc h as the rate of transition or absorption cross section ma y b e calculated from V k m : Additionally time-dep enden t p erturbation theory giv es a prescription for calculating successiv ely higher order corrections to V k m ; whic h ma y in turn pro vide b etter and b etter agreemen t with exp erimen t. This section concludes with a discussion of quan tum electro dynamics, in whic h b oth matter and elds are quan tized and the description is relativistically in v arian t. Here again the time-dep enden t p erturbation theory (often in the form of F eynman diagrams) is the essen tial mac hinery used in calculations. 3.1.1 Time-Dep enden t P erturbation Theory An imp ortan t class of solutions to the Sc hr odinger equation ( 1.1 ) are those whic h are eigenfunctions of the Hamiltonian op erator H : These solutions j m i satisfy the time-indep enden t Sc hr odinger equation H j m i = E m j m i (3.1) and are called stationary states. A general solution j ( t ) i of the Sc hr odinger equation ( 1.1 ) ma y b e constructed from these stationary states according to j ( t ) i = e iH ( t t 0 ) j ( t 0 ) i = P m e iE m ( t t 0 ) j m ih m j ( t 0 ) i ; (3.2)

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31 where j ( t 0 ) i is an initial state v ector and where the sum o v er m ma y imply in tegration if the energy sp ectrum is con tin uous. Equation ( 3.2 ) is only applicable when the Hamiltonian is time-indep enden t. F or if H H ( t ) ; then the energy of the system is not conserv ed and H admits no strictly stationary states. Ho w ev er, it ma y b e p ossible to split a time-dep enden t Hamiltonian in to the sum of t w o terms: H = H 0 + V ( t ) ; (3.3) where H 0 is time-indep enden t and describ es the unp erturb ed system while V ( t ) accoun ts for the time-dep enden t p erturbation. T o x ideas, consider for example the electronic transition induced b y a passing electromagnetic disturbance that is lo calized in b oth space and time. In other w ords, the system is initially unp erturb ed for some long time and is in an eigenstate of H 0 : While in the in teraction region the system is p erturb ed b y V after whic h it settles do wn in to another unp erturb ed eigenstate of H 0 for a sucien tly long time. Time-dep enden t p erturbation theory seeks to connect the stationary states of the unp erturb ed system, i.e. those states satisfying H 0 j m i = E m j m i ; (3.4) with the time-dep enden t p erturbation V ( t ) : These calculations are most clearly demonstrated in the in teraction picture. In the in teraction picture the p erturbation is singled out b y applying the unitary op erator U 0 = exp ( iH 0 t ) to j ( t ) i : That is j I ( t ) i = e iH 0 t j ( t ) i (3.5) and the time-dep enden t Sc hr odinger equation ( 1.1 ) b ecomes i ( d=dt ) j I ( t ) i = V I ( t ) j I ( t ) i ; (3.6)

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32 where V I ( t ) = U 0 V ( t ) U y 0 : In other w ords, the in teraction picture separates the ph ysics that dep ends up on the p erturbation from the ph ysics that dep ends up on only the unp erturb ed system. The state v ector at time t is obtained from that at time t 0 via j I ( t ) i = U I ( t; t 0 ) j I ( t 0 ) i ; (3.7) where U I is the time ev olution op erator whic h satises U I ( t; t 0 ) = 1 i R tt 0 dt 0 V I ( t 0 ) U I ( t 0 ; t 0 ) : (3.8) The time ev olution op erator connects the (orthonormal) stationary states j k i and j m i according to h k j U I ( t; t 0 ) j m i = k m i R tt 0 dt 0 h k j V I ( t 0 ) U I ( t 0 ; t 0 ) j m i = k m i P n R tt 0 dt 0 h k j V I ( t 0 ) j n ih n j U I ( t 0 ; t 0 ) j m i = k m i P n R tt 0 dt 0 e i! k n t 0 h k j V ( t 0 ) j n ih n j U I ( t 0 ; t 0 ) j m i ; (3.9) where P n j n ih n j = 1 and k n = E k E n w ere used. The time-dep enden t p erturbation theory is no w set up b y iterating on ( 3.8 ). If the p erturbation V is small then the time ev olution op erator b ecomes a p o w er series in V : That is U I ( t; t 0 ) = 1 i R tt 0 dt 0 V I ( t 0 ) + ( i ) 2 R tt 0 dt 0 V I ( t 0 ) R t 0 t 0 dt 00 V I ( t 00 ) + : (3.10) And so at rst order the transition amplitude b et w een t w o distinct states of energy E k and E m (with k 6 = m ) is h k j U I ( t; t 0 ) j m i = i R tt 0 dt 0 e i! k m t 0 h k j V ( t 0 ) j m i : (3.11)

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33 Assuming that the p erturbation is sucien tly small, the probabilit y of nding the system in the state j k i is giv en b y P k m ( t ) = h k j U I ( t; t 0 ) j m i 2 = i R tt 0 dt 0 e i! k m t 0 h k j V ( t 0 ) j m i 2 : (3.12) If the p erturbation is lo calized in time then t 0 and t ma y b e naiv ely extended to innit y to yield the transition probabilit y P k m (+ 1 ) = h k j U I (+ 1 ; 1 ) j m i 2 = i R 11 dt e i! k m t h k j V ( t ) j m i 2 (3.13) whic h in v olv es a F ourier in tegral of the matrix elemen t V k m = h k j V ( t ) j m i : 3.1.2 F ermi Golden Rule The formalism set up th us far is also applicable for time-indep enden t p erturbations V 6 = V ( t ) : In this case the transition probabilit y can b e obtained from ( 3.12 ) as P k m ( t ) = 2 jh k j V j m ij 2 1 cos ( k m t ) ( E k E m ) 2 (3.14) whic h is prop ortional to t 2 if E k E m : No w consider the situation in whic h there is a near con tin uum of nal states a v ailable ha ving energies in the in terv al ( E m E = 2 ; E m + E = 2) : If the densit y of the near con tin uum states is denoted b y F ( E ) ; then the transition probabilit y to all of these states is giv en b y P k 2 F P k m ( t ) = R E m + E = 2 E m E = 2 2 jh k j V j m ij 2 1 cos( k m t ) ( E k E m ) 2 F ( E k ) dE k ; (3.15) where the sum runs o v er all states j k i b elonging to the near con tin uum of nal states. The quotien t [1 cos ( k m t )] = ( E k E m ) 2 is sharply p eak ed at E k = E m whic h conrms that the dominan t transitions are those that conserv e the unp erturb ed energy Since b oth jh k j V j m ij 2 and F ( E k ) are appro ximately constan t around E m and t is suc h that E 2 =t ( i.e. long time b eha vior), the transition probabilit y

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34 b ecomes P k 2 F P k m ( t ) 2 jh k j V j m ij 2 F ( E k ) R 11 1 cos ( k m t ) 2 k m d! k m = 2 t jh k j V j m ij 2 F ( E k ) (3.16) whic h increases linearly with time. The total transition probabilit y p er unit time or tr ansition r ate is giv en b y = ( d=dt ) P k 2 F P k m ( t ) = 2 jh k j V j m ij 2 F ( E k ) (3.17) and is constan t. F ermi's golden rule of time-dep enden t p erturbation theory [ 7 34 35 ] em b o dies the tendency for the p erturb ed system to mak e energy conserving transitions for whic h the probabilit y increase as t 2 or to mak e nearly energy conserving transitions whic h oscillate in time. Either w a y the transition rate is constan t. F ermi's golden rule ma y b e extended to include p erturbations that v ary harmonically in frequency : An electromagnetic disturbance of a c harge w ould b e an example. In this case the golden rule generalizes to = 2 jh k j V j m ij 2 F ( E m + ) : (3.18) 3.1.3 Absorption of Electromagnetic Radiation b y an A tom Recall the electromagnetic eld coupling to quan tum mec hanics is giv en b y the minimal c oupling prescription i ( @ =@ t ) i ( @ =@ t ) q i r i r q A =c; (3.19) where A = ( ; A ) are the dynamical v ariables of the electromagnetic eld. Applying this transformation to the Sc hr odinger equation i = P 2 = 2 m + V 0 results in the Sc hr odinger equation coupled to the electromagnetic eld i = [ P q A =c ] 2 2 m + V 0 + q (3.20)

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35 with Hamiltonian H = [ P q A =c ] 2 2 m + V 0 + q : (3.21) In the Coulom b gauge (see App endix A for details) this Hamiltonian b ecomes H = P 2 2 m + V 0 q mc A P + q 2 2 mc 2 A 2 : (3.22) The external free electromagnetic eld ev olv es according to r 2 A A =c 2 = 0 with r A = 0 and = 0 since it is assumed that the c harges do not themselv es con tribute to the eld. By ignoring the quadratic term in A ; the Hamiltonian H separates in to an unp erturb ed p ortion H 0 = P 2 2 m + V 0 (3.23) plus the p erturbation V = q mc A P : (3.24) It should b e p oin ted out here that substan tial confusion has existed in the literature o v er the A P app earing in the p erturbation V : This confusion w as due the widespread use of E r and its higher order appro ximations [ 59 60 ] instead of A P : The relationship b et w een these t w o approac hes ha v e b een thoroughly in v estigated in [ 76 { 79 ]. The cross section for stim ulated absorption (or emission) of radiation b y an atom ma y b e calculated via F ermi's golden rule. If the external eld v aries harmonically in frequency as a plane w a v e, then the p erturbation b ecomes V ( x ; t ) = q mc A 0 e i ( k x t ) + A 0 e i ( k x t ) ^ P (3.25) where ^ is the eld's p olarization. The rate of energy absorption b y the atom is = q 2 m 2 c 2 j A 0 j 2 h k j e i k x ^ P j m i 2 F ( E m + ) : (3.26)

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36 If the densit y of the near con tin uum states is narro w then F ( E m + ) = ( E k E m + ) = ( k m + ) (3.27) and the absorption cr oss se ction abs ( ) = =I 0 b ecomes abs ( ) = q 2 j A 0 j 2 h k j e i k x ^ P j m i 2 ( k m + ) =m 2 c 2 j A 0 j 2 = 2 c = 4 2 q 2 m 2 c j A 0 j 2 h k j e i k x ^ P j m i 2 ( k m + ) ; (3.28) where I 0 = j A 0 j 2 = 2 c is the inciden t rux of photons of frequency : Similarly the emission cross section is em ( ) = 4 2 q 2 m 2 c j A 0 j 2 h k j e i k x ^ P j m i 2 ( k m ) : (3.29) Notice that the time-dep enden t p erturbation theory giv es prop erties of the solution but not the solution. That is, the cross section is easily accessible but the w a v efunction and 4-p oten tial are not. The cross section is a prop ert y of the solution and can b e calculated from kno wledge of the solution. Of course, the w a v efunction and 4-p oten tial constitute the actual solution. 3.1.4 Quan tum Electro dynamics in Brief The quan tum theory of electro dynamics [ 19 24 30 ], also kno wn as QED, is the in teracting quan tum eld theory of electron and photon elds. The relativistically in v arian t QED is one of the most successful ph ysical theories to date, in that there is no evidence for an y discrepancy b et w een exp erimen t and prediction. Ho w ev er, QED is b eset b y man y mathematical and logical diculties. These diculties are in some cases a v oided b y ph ysical argumen ts or simply concealed from view as in the renormalization of mass and c harge. Putting aside its inconsistencies, QED is a prime example of the success of time-dep enden t p erturbation theory A com bination of the free Dirac theory and the free Maxw ell theory pro vide the unp erturb ed states on whic h the in teraction

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37 L in t = J A =c op erates. The free QED Lagrangian densit y L free QED = [ ir @ mc ] 1 16 F F (3.30) giv es the equations of motion for the free electron [ ir @ mc ] = 0 (3.31) and the equations of motion for the free electromagnetic eld @ F = @ 2 A @ ( @ A ) = 0 ; (3.32) where the Dirac r -matrices are related to the P auli spin matrices ( 4.66 ), = y r 0 is the Dirac adjoin t of the four comp onen t spinor ; and F = @ A @ A is the electromagnetic eld tensor. This nonin teracting theory sets up the free unp erturb ed in-state j p 1 p n i in and out-state j k 1 k m i out ; whic h will b e connected b y U I (+ 1 ; 1 ) = T f exp [ i R d 4 x L in t ] g ; where T is the time-ordering op erator. The resulting matrix elemen ts will yield some prop erties of the dynamics. W orking in the in teraction picture, the mac hinery of time-dep enden t p erturbation theory is used to construct the sc attering matrix or S-matrix out h k 1 k m j p 1 p n i in = in h n j a ( k 1 ) a ( k m ) U I (+ 1 ; 1 ) a y ( p 1 ) a y ( p n ) j n i in (3.33) where S = 1 + iT and j n i in is the in-v acuum. The resp ectiv e fermion cr e ation and annihilation op er ators a y ( k ) and a ( k ) ; create and annihilate single fermions of momen tum k according to a y ( k ) j n i = j k i and a ( k ) j n i = 0 where the spin has b een neglected for simplicit y The situation in whic h the particles do not in teract at all (the 1 part) as w ell as the in teresting in teractions (the iT part) are b oth included in the S -matrix. The in teracting comp onen ts are commonly collected and are referred

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38 to as the T-matrix. T ogether 1 + iT is used to dene the invariant amplitude M as out h k 1 k m j U I (+ 1 ; 1 ) j p 1 p n i in = (2 ) 4 ( p 1 + + p n k 1 k m ) i M ( p 1 ; ; p n k 1 ; ; k m ) : (3.34) This in v arian t amplitude is analogous to the scattered w a v efunction of quan tum mec hanics, i.e. k ( x ) N h e i k x + f k (n) e ik r r i ; (3.35) where the unscattered eld exp ( i k x ) and the spherically scattered eld exp ( ik r ) =r are indicated sc hematically in Figure 3{1 In fact all of quan tum mec hanics is PSfrag replacemen ts Scattered eld Inciden t eld Figure 3{1: The co ecien t 1 of the unscattered plane w a v e exp ( i k x ) is analogous to the 1 part of the S -matrix, while the scattering amplitude f k (n) whic h mo dulates the scattered spherical w a v e exp ( ik r ) =r is analogous to the iT part. just the nonrelativistic limit of QED. Unfortunately while QED is suitable for the scattering of single particle states to single particle states, it requires great eort to deal with b ound states. The probabilit y of nding j k 1 k m i out in j p 1 p n i in is giv en b y P (+ 1 ) out h k 1 k m j p 1 p n i in 2 d 3 k 1 (2 ) 3 d 3 k m (2 ) 3 = in h n j a ( k 1 ) a ( k m ) U I (+ 1 ; 1 ) a y ( p 1 ) a y ( p n ) j n i in 2 d 3 k 1 (2 ) 3 d 3 k m (2 ) 3 (3.36)

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39 whic h is analogous to ( 3.13 ). A similar connection can b e made in the cross section. If n = 2 in the in-state, then the dieren tial scattering cross section d b ecomes d i M ( p 1 ; p 2 k 1 ; ; k m ) 2 d 3 k 1 (2 ) 3 d 3 k m (2 ) 3 (2 ) 4 ( p 1 + p 2 k 1 k m ) (3.37) whic h is analogous to ( 3.28 ). As in quan tum mec hanics, time-dep enden t p erturbation theory in QED giv es a prescription b y whic h to calculate prop erties of the solution whic h rely on scattering amplitudes, e.g. cross sections, deca y rates, and probabilities. It is considerably more dicult to compute the actual solution, whic h in this case w ould b e the states on whic h the eld op erators A ( x ; t ) = R V d 3 k (2 ) 3 P n ( k ; ) c ( k ; ) e i k x p 2 k + ( k ; ) c y ( k ; ) e i k x p 2 k o i ( x ; t ) = R V d 3 k (2 ) 3 P s n u i ( k ; s ) a ( k ; s ) e i k x p 2 + v i ( k ; s ) b y ( k ; s ) e i k x p 2 o (3.38) act. In ( 3.38 ), and are helicit y eigenstates of A ; and f c; c y g are photon creation and annihilation op erators. Similarly u i and v i are eigenspinors of i ; and f a; a y g and f b; b y g are electron and p ositron [ 80 ] creation and annihilation op erators. Lastly it should b e p oin ted out that a b eautiful represen tation of the timedep enden t p erturbation theory w as in tro duced b y F eynman [ 55 ]. These so called F eynman diagr ams pro vide a pictorial v ersion of the in v arian t amplitude i M = C out h k 1 k m j U I (+ 1 ; 1 ) j p 1 p n i in = C out h k 1 k m j T n e i R d 4 x L in t o j p 1 p n i in (3.39) = h out h k 1 k m j 1 + i R d 4 x T L in t + j p 1 p n i in i connected :

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40 F or example, the in v arian t amplitude for Bhabha scattering, i.e. e + e e + e ; is i M Bhabha = h out h k 1 k 2 j p 1 p 2 i in + out h k 1 k 2 j i R d 4 x T L in t j p 1 p 2 i in + O ( L 2in t ) i connected = p 1 p 2 k 1 k 2 + p 1 p 2 k 1 k 2 + O ( L 4in t ) ; (3.40) where eac h of the ab o v e diagrams corresp onds to a term (or p ortion thereof ) in the p erturbativ e expansion of i M Bhabha : These tree order diagrams are the lo w est order non v anishing diagrams that con tribute to and are the largest part of the Bhabha scattering in v arian t amplitude. Higher order p erturbativ e corrections to the amplitude also ha v e pictorial represen tations and ma y b e systematically constructed using F eynman's prescription. In this manner the time-dep enden t p erturbation theory ma y b e diagrammatically written to an y order, translated in to mathematical expressions, and computed. While this is b y no means an easy task, the in v arian t amplitude ma y in principle b e calculated to an y order. Notice again that this mac hinery pro duces the amplitude i M ; whic h is a prop ert y of the solution but not the actual solution. 3.2 Classical Electro dynamics Sp ecied b y the Sources and J If the sources of c harge and curren t are kno wn, then the dynamics of the resulting electromagnetic eld can b e calculated from Maxw ell's equations at eac h p oin t in space-time. These elds ma y b eha v e quite dieren tly dep ending on the motion of their source. F or example, a static source giv es rise to a purely electrostatic eld, while a uniformly mo ving source creates b oth an electric eld and a magnetic eld. More imp ortan tly if the source is accelerated then ele ctr omagnetic r adiation is produced. Electromagnetic radiation is a unique kind of electromagnetic eld in that

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41 it carries a w a y energy momen tum, and angular momen tum from its source. The radiation eld is not b ound to the c harge as are the static elds. In this section the electromagnetic elds pro duced b y an oscillating electric dip ole are calculated to lo w est order via the m ultip ole expansion. The corresp onding p o w er and radiation pattern are also presen ted. Then, the dynamics of the bremsstrahlung pro duced b y a w a v epac k et source is analyzed. It is sho wn that the w a v epac k et's elds reduce in the narro w width limit to the usual p oin t source results. The consisten t coupling of electro dynamics and quan tum mec hanics is needed b ecause the sources of c harge and curren t pro duce electromagnetic elds and these elds act bac k up on the sources. The understanding of this pro cess requires the inclusion of recoil eects on the c harges due to the electromagnetic eld. These eects, kno wn as r adiation r e action eects, are a main asp ect of this dissertation and will b e discussed in detail in Chapter 4 3.2.1 Electromagnetic Radiation from an Oscillating Source In this section the Lorenz gauge (see App endix A for details) is used to in v estigate the electromagnetic radiation pro duced b y a lo calized system of c harge and curren t [ 9 ] whic h v ary sin usoidally in time according to (the real part of ): ( x ; t ) = ( x ) e i! t J ( x ; t ) = J ( x ) e i! t : (3.41) It is assumed that the electomagnetic p oten tials and elds also ha v e the same timedep endence. The general solutions to the w a v e equations of ( A.5 ) are giv en b y ( x ; t ) = R V d 3 x 0 R 11 dt 0 ( x 0 ; t 0 ) j x x 0 j t 0 h t + j x x 0 j c i A ( x ; t ) = 1 c R V d 3 x 0 R 11 dt 0 J ( x 0 ; t 0 ) j x x 0 j t 0 h t + j x x 0 j c i ; (3.42) where G (+) ( x ; t ; x 0 ; t 0 ) = ( t 0 [ t + j x x 0 j =c ]) = 4 j x x 0 j is the retarded Green's function for the w a v e op erator @ 2 = = @ 2 =@ ( ct ) 2 r 2 (see App endix B ). It is

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42 assumed that there are no b oundary surfaces presen t. With the oscillating sources from ( 3.41 ), it will b e seen that all of the dynamics of the electromagnetic eld for whic h 6 = 0 can b e describ ed in terms of the A alone. The comp onen t of the electromagnetic eld for whic h = 0 is just the static electric monop ole eld monop ole ( x ; t ) = q j x j : (3.43) The v ector p oten tial for all other frequencies is A ( x ) = 1 c R V d 3 x 0 e ik j x x 0 j j x x 0 j J ( x 0 ) ; (3.44) where the w a v ev ector k = =c and it is understo o d that A ( x ; t ) = A ( x ) e i! t : F or a giv en c harge densit y J ; ( 3.44 ) could in principle b e computed. With the resulting v ector p oten tial the electromagnetic eld ma y b e calculated from Amp ere's la w. That is B = r A (3.45a) E = i k r B (3.45b) in a region outside the source. Instead of ev aluating ( 3.44 ) exactly general prop erties of its solution ma y b e determined whenev er the dimensions of J are m uc h smaller than a w a v elength. That is, if the dimensions of the c harge densit y are of order d and the w a v elength = 2 =k ; then d : F rom these distances, the follo wing three spatial regions ma y b e constructed: The near or static zone: d r The in termediate zone: d r The far or r adiation zone: d r In eac h region the electromagnetic eld b eha v es quite dieren tly F or example in the near zone, the elds b eha v e as if they w ere static elds whic h sho w strong

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43 dep endence on their source. On the other hand in the far zone, the elds displa y prop erties of radiation elds whic h are transv erse and fall o as r 1 : The static near zone elds ma y b e obtained from ( 3.44 ) b y noting that k r 1 since r : In this case exp ( ik j x x 0 j ) 1 and the v ector p oten tial b ecomes A near ( x ) = 1 c R V d 3 x 0 J ( x 0 ) j x x 0 j = 1 c R V d 3 x 0 J ( x 0 ) n P l m 4 2 l + 1 r 0 l r l +1 Y l m (n) Y l m (n 0 ) o = ik p r + ; (3.46) where j x x 0 j 1 has b een expanded in to the spherical harmonics Y l m and an in tegration b y parts w as p erformed with all surface terms v anishing. The equation of con tin uit y i! + r J = 0 w as also used in the computation as w ell as the denition of the dip ole momen t p = R V d 3 x x ( x ) : F rom ( 3.45 ), the resulting magnetic and electric elds are: B near = ik r 2 ^ n p + (3.47a) E near = 3 ^ n ( ^ n p ) p r 3 + ; (3.47b) where ^ n is the unit v ector in the direction of the observ ation p oin t x : Notice that E near is indep enden t of the frequency and is th us purely static. As exp ected B near is zero in the static limit ! 0 : A m ultip ole expansion of the near zone v ector p oten tial can no w b e made and successiv ely b etter results ma y b e obtained b y going to higher orders in ( l ; m ) :

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44 A t the other extreme, the far zone elds for whic h k r 1 ma y b e obtained from ( 3.44 ) b y noticing that j x x 0 j = p ( x x 0 ) ( x x 0 ) = j x j s 1 2 x x 0 j x j 2 + j x 0 j 2 j x j 2 j x j 1 x x 0 j x j 2 = r ^ n x 0 (3.48) since j x 0 j j x j = r j ^ n j = r : A picture of the corresp onding situation is sho wn in Figure 3{2 where the x 0 -in tegration runs o v er the domain of the source J : With the PSfrag replacemen ts d O ^ n r = j x j x Figure 3{2: In the radiation zone, the observ ation p oin t x is lo cated far from the source J : In this case the distance j x x 0 j r ^ n x 0 : appro ximation ( 3.48 ), the far zone v ector p oten tial b ecomes A far ( x ) = e ik r r 1 c R V d 3 x 0 J ( x 0 ) e ik ^ n x 0 = e ik r r 1 X m =0 ( ik ) m m 1 c R V d 3 x 0 J ( x 0 )( ^ n x 0 ) m = ik p e ik r r + ; (3.49) where j x x 0 j 1 r 1 if only the leading term in k r is k ept. It can no w b e seen that the v ector p oten tial is an outgoing spherical w a v e with m th-angular co ecien t R V d 3 x 0 J ( x 0 )( ik ^ n x 0 ) m =cm : F rom ( 3.45 ), the corresp onding elds are: B far = k 2 e ik r r ( ^ n p ) h 1 1 ik r i + (3.50a) E far = k 2 e ik r r ( ^ n p ) ^ n + h 1 r 3 ik r 2 i e ik r [3 ^ n ( ^ n p ) p ] + : (3.50b)

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45 The magnetic eld is transv erse to the radius v ector x = r ^ n while the electric eld has comp onen ts longitudinal and transv erse to x : Both elds fall o lik e r 1 at leading order. The r 1 -elds are the true radiation elds whic h carry energy momen tum, and angular momen tum to innit y This can b e seen from the timea v eraged dier ential p ower radiated p er unit solid angle dP d n = 1 2 Re h r 2 ^ n c 4 E B i = ck 4 8 j ^ n p j 2 = ck 4 8 j p j 2 sin 2 (3.51) whic h in this case is a measure of the energy radiated p er unit time p er unit solid angle b y an oscillating electric dip ole p : In tegrating this expression o v er n = ( ; ) giv es the total p o w er radiated, i.e. P = R d n dP d n = ck 4 3 j p j 2 : (3.52) The corresp onding radiation pattern is sho wn in Figure 3{3 In general, the p o w er radiated b y an l -p ole go es lik e k 2( l +1) : Notice that it is the r 1 -elds whose p o w er mak es it to innit y in three dimensions. This is b ecause E B r 2 whic h exactly cancels the r 2 in the measure factor d 3 x = r 2 dr d n : In t w o dimensions, it is the r 1 = 2 -elds whose p o w er mak es it to innit y since d 2 x = r dr d : As b efore, a more accurate description of the radiation eld is obtained b y including higher order terms in the sum ( 3.49 ). The lo w est order (non v anishing) m ultip ole con tributes the most to the eld. In the in termediate zone, neither of the previous appro ximations are v alid. In fact all terms in the previous series expansions w ould ha v e to b e k ept. The understanding of the b eha vior of the elds in this zone requires the more sophisticated mac hinery of v ector m ultip ole elds. The in terested reader is referred to [ 9 ] for a detailed discussion of m ultip ole elds of arbitrary order ( l ; m ) :

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46 p xFigure 3{3: The dieren tial p o w er dP =d n or radiation pattern corresp onding to an oscillating electric dip ole v eries that no radiation is emitted in the direction of the dip ole momen t. Rather the dip ole radiation is a maxim um in the direction transv erse to p : Outside of the ph ysics literature there is also a large amoun t of engineering literature in the eld of computational electro dynamics. In this area, Maxw ell's eld equations are often solv ed n umerically b y nite elemen t metho ds (see Jiao and Jin [ 81 ] and references therein). Man y applications of this w ork lie in electromagnetic scattering, w a v eguiding, and an tenna design. The in v erse source problem [ 82 ] is also another area of in terest in engineering. Here, the goal is to determine the sources of c harge and curren t with only the kno wledge of the electromagnetic elds outside of the source's region of supp ort. This problem has b eneted from the w ork of Go edec k e [ 83 ], Dev aney and W olf [ 84 ], Marengo and Ziolk o wski [ 85 ], and Ho enders and F erw erda [ 86 ], who ha v e demonstrated the decomp osition of the electromagnetic eld in to nonradiating and purely radiating comp onen ts.

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47 3.2.2 Electromagnetic Radiation from a Gaussian W a v epac k et Consider the gaussian w a v epac k et with initial p osition r mo ving with constan t v elo cit y v ( x ; t ) = h 2 ` 2 i 3 = 4 e ` 2 [ x ( r + v t )] 2 e im v x ; (3.53) where b = 1 = p 2 ` is the w a v epac k et width. The corresp onding probabilit y curren t is giv en b y J ( x ; t ) = v ( x ; t ) = q 2 m [ ( i r ) + ( i r )] : (3.54) In F ourier space this curren t b ecomes ~ J ( k ; t ) = q v exp ( i k [ r + v t ] k 2 = 8 ` 2 ) and the v ector p oten tial is obtained b y in tegrating against the Green's function D (+) k for the w a v e op erator (see App endix B ). The v ector p oten tial b ecomes A ( x ; t ) = R V d 3 k (2 ) 3 e i k x R 11 dt 0 D (+) k ( t; t 0 )4 c ~ J ( k ; t 0 ) = 4 cq v R V d 3 k (2 ) 3 e i k ( x r ) k 2 = 8 ` 2 R 11 dt 0 ( t t 0 ) sin ck ( t t 0 ) ck e i k v t 0 = 4 cq v R V d 3 k (2 ) 3 e i k [ x ( r + v t )] k 2 = 8 ` 2 c 2 k 2 ( k v ) 2 (3.55) whic h is dicult to p erform analytically due to the complicated angular dep endence of the in tegrand. F or nonrelativistic v elo cities, A can b e appro ximated b y A ( x ; t ) = 4 cq v R V d 3 k (2 ) 3 e i k [ x ( r + v t )] k 2 = 8 ` 2 c 2 k 2 [1 ( v =c ) 2 cos 2 ] q v 2 2 c R V d 3 k k 2 e i k [ x ( r + v t )] k 2 = 8 ` 2 = q v c erf p 2 ` j x ( r + v t ) j j x ( r + v t ) j ; (3.56) where Gradsh teyn and Ryzhik [ 87 ] w as used. The norms of this v ector p oten tial and its asso ciated curren t densit y J are plotted along the x -axis in Figure 3{4 for t w o dieren t v elo cities. The c harge q is tak en to b e negativ e. Notice that A follo ws the c harge distribution and that A will generate an electromagnetic eld. F or v =c 1 this result is equiv alen t to a Galilei b o ost of the elds from the rest frame of the

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48 x 20 15 10 5 0 -5 1 0.5 0 Figure 3{4: The norms of J and A are plotted with dieren t v elo cities along the x -axis. source. Only the electrostatic eld remains b y going to the rest frame. And so, there is little dierence b et w een uniform motion and no motion. As stated previously the more in teresting eld dynamics is created whenev er the source is accelerated. T o this end, consider the v ector p oten tial arising from a mo ving c harge whose curren t has the simple time dep endence ~ J ( k ; t ) = q v ( t ) e i k ( r + v t ) k 2 = 8 ` 2 = q v e i k ( r + v t ) k 2 = 8 ` 2 ( t t 0 )( t 1 t ) ; (3.57) where v is constan t. This time dep endence corresp onds to a situation in whic h a source is suddenly accelerated from a standstill to a uniform mo v emen t with v elo cit y v and is then instan taneously decelerated again to a standstill (see Figure 3{5 ). In eac h of the three temp oral regions of the curren t, the v ector p oten tial has a dieren t

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49 PSfrag replacemen ts r ( t ) ct x past elsewhere future t 1 t 2 Figure 3{5: The tra jectory or w orld line r ( t ) of the c harge is plotted. Electromagnetic radiation is pro duced at t 1 and t 2 and mo v es out on the ligh t cone. b eha vior. Ob viously for t < t 0 ; A ( x ; t ) = 0 : F or t 0 t < t 1 ; A ( x ; t ) q v c n erf p 2 ` j x ( r + v t ) j j x ( r + v t ) j + 1 2 erf p 2 ` [ c ( t t 0 ) j x ( r + v t 0 ) j ] j x ( r + v t 0 ) j 1 2 erf p 2 ` [ c ( t t 0 ) + j x ( r + v t 0 ) j ] j x ( r + v t 0 ) j o (3.58) and for t t 1 ; A ( x ; t ) q v 2 c n erf p 2 ` [ c ( t t 0 ) j x ( r + v t 0 ) j ] j x ( r + v t 0 ) j erf p 2 ` [ c ( t t 0 ) + j x ( r + v t 0 ) j ] j x ( r + v t 0 ) j + erf p 2 ` [ c ( t t 1 ) + j x ( r + v t 1 ) j ] j x ( r + v t 1 ) j erf p 2 ` [ c ( t t 1 ) j x ( r + v t 1 ) j ] j x ( r + v t 1 ) j o : (3.59) Again nonrelativistic v elo cities are assumed. A space-time plot of the norm of this piecewise v ector p oten tial is sho wn in Figure 3{6 Note that the c harge w as at rest un til the time t 0 ; where it w as instan taneously accelerated to a v elo cit y of magnitude

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50 PSfrag replacemen ts 10 10 -10 0 0 0 5 5 -5 2 -2 t x Figure 3{6: The bremsstrahlung radiation from a c harged gaussian w a v epac k et mo v es out on the smeared ligh t cone with maxim um at x = ct: v : Then the c harge mo v ed uniformly with v un til the time t 1 ; when it w as instantaneously decelerated to rest again. Since electromagnetic radiation is pro duced whenev er the v elo cit y c hanges in time, electromagnetic ripples are pro duced at t 0 and t 1 : The ripples mo v e out as radiation at the v elo cit y c of ligh t. Figure 3{6 sho ws the ligh t cone, whic h is smeared out due to the nonp oin tlik e structure of the c harge. The v ector p oten tial presen ted so far has b oth longitudinal and transv erse comp onen ts. F or the time b eing, the tran v ersalit y of the A is not imp ortan t. It turns out that the only elds whic h con tribute to the P o yn ting v ector or to the p o w er are the transv erse elds. And so it do es no harm to k eep the full v ector p oten tial. F or the in terested reader, the transv erse v ector p oten tial A T asso ciated with ( 3.58 ) and ( 3.59 ) is calculated in App endix C b y analogy to the quadrup ole momen t tensor.

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51 The electric and magnetic elds corresp onding to ( 3.59 ) are E A c = q v ` c r 2 h e 2 ` 2 [ c ( t t 0 ) R ( t 0 )] 2 e 2 ` 2 [ c ( t t 0 )+ R ( t 0 )] 2 R ( t 0 ) + e 2 ` 2 [ c ( t t 1 )+ R ( t 1 )] 2 e 2 ` 2 [ c ( t t 1 ) R ( t 1 )] 2 R ( t 1 ) i = q v ` c r 2 g 0 ( t ) + g 1 ( t ) | {z } h ( t ) (3.60) neglecting the purely longitudinal r ; and B = r A = v q ` c r 2 h ^ u ( t 1 ) e 2 ` 2 [ c ( t t 1 )+ R ( t 1 )] 2 + e 2 ` 2 [ c ( t t 1 ) R ( t 1 )] 2 R ( t 1 ) ^ u ( t 0 ) e 2 ` 2 [ c ( t t 0 ) R ( t 0 )] 2 + e 2 ` 2 [ c ( t t 0 )+ R ( t 0 )] 2 R ( t 0 ) i = q ` c r 2 v h ^ u ( t 1 ) g + 1 ( t ) ^ u ( t 0 ) g + 0 ( t ) i ; (3.61) where R ( t ) j R ( t ) j = j x ( r + v t ) j and where the unit v ectors ^ u ( t 0 ) = R ( t 0 ) =R ( t 0 ) and ^ u ( t 1 ) = R ( t 1 ) =R ( t 1 ) : With ( 3.60 ) and ( 3.61 ), the Poynting ve ctor is S = c 4 E B = q 2 ` 2 2 2 c h ( t ) v n v h ^ u ( t 1 ) g + 1 ( t ) ^ u ( t 0 ) g + 0 ( t ) io : (3.62) The dieren tial p o w er radiated in to the solid angle d n at time t b ecomes dP ( x ; t ) d n = R ( t ) 2 ^ n S ( t ) = q 2 ` 2 2 2 c h ( t ) R ( t ) 2 nh v ^ u ( t 1 ) g + 1 ( t ) v ^ u ( t 0 ) g + 0 ( t ) i ^ n v v 2 h ^ n ^ u ( t 1 ) g + 1 ( t ) ^ n ^ u ( t 0 ) g + 0 ( t ) io = q 2 ` 2 2 2 c h ( t ) R ( t ) 2 n g + 0 ( t ) h v ^ n v ^ u ( t 0 ) i g + 1 ( t ) h v ^ n v ^ u ( t 1 ) io (3.63)

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52 where the unit v ector ^ n = R ( t ) =R ( t ) is normal to the surface of the ball that emanates from the radiation source. The v ectors v ; ^ n ; ^ u ( t 0 ) ; and ^ u ( t 1 ) are all constan t in time. By c ho osing the z -axis along the v elo cit y v ; the angles b et w een v and the unit v ectors ^ u ( t 0 ) and ^ u ( t 1 ) are 0 and 1 resp ectiv ely With a little geometry it can b e v eried that sin 0 sin p 1 + ( v =ct )( t 1 t 0 ) cos and sin 1 sin p 1 ( v =ct )( t 1 t 0 ) cos (3.64) b y suppressing terms of quadratic order and higher in c 1 where is the angle b et w een v and ^ n : In terms of the angles 0 ; 1 ; and ; the dieren tial p o w er b ecomes dP d n = q 2 ` 2 v 2 2 2 c h ( t ) R ( t ) 2 h g + 0 ( t ) sin 0 sin g + 1 ( t ) sin 1 sin i (3.65) whic h is indep enden t of the p olar angle : The corresp onding radiation pattern is sho wn in Figure 3{7 and sho ws that p o w er is radiated in all directions except along the direction of motion. Notice that the \dip ole-lik e" pattern is mo died b y con tributions arising from the expansion of the square ro ots in the angles 0 and 1 : That is sin 2 p 1 ( v =ct )( t 1 t 0 ) cos = sin 2 h 1 v 2 ct ( t 1 t 0 ) cos + O ( v =c ) 2 cos 2 i : (3.66) These con tributions are more signican t at higher v elo cities. The quadrup ole pattern in Figure 3{7 is ob viously o v eremphasized. By in tegration o v er the unit sphere, the total p o w er is found to b e P ( x ; t ) = q 2 ` 2 v 2 c h ( t ) R ( t ) 2 R 0 d n g + 0 ( t ) sin 0 sin 2 g + 1 ( t ) sin 1 sin 2 o (3.67) whic h is equiv alen t to dE =dt where E is the total eld energy Both of the in tegrals in ( 3.67 ) can b e done analytically Since b oth h and g + are prop ortional to 1 =R ; the p o w er do es not deca y with the radius x :

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53 PSfrag replacemen ts x v Figure 3{7: The radiation pattern giv en b y ( 3.63 ) sho ws the c haracteristic dip ole pattern at lo w est order. Keeping O ( v =c ) terms rev eals the quadrup ole pattern. Higher order m ultip ole patterns are generated b y O ( v 2 =c 2 ) and higher terms. F or an electron whose c harge distribution has a width corresp onding to the Bohr radius a 0 and has a v elo cit y of v e = 1 a.u. b et w een the times t 0 = 0 a.u. and t 1 = 1 a.u. ; the instan taneous p o w er is P 2 10 3 a.u. 3 10 4 J/s at the maxim um of the p eak from t 0 : The p o w er from the t 1 p eak is the same. In order to put the previous results in to p ersp ectiv e it is useful to mak e a comparison with the Larmor result. The L armor p ower P ( t ) = 2 q 2 3 c 3 v ( t ) 2 (3.68) is the instan taneous p o w er radiated b y an accelerated p oin t c harge that is observ ed in a reference frame where the v elo cit y of the c harge is signican tly less than that of ligh t. The angular b eha vior of the emitted radiation ma y b e determined b y examining the dieren tial p o w er dP ( t ) d n = q 2 4 c 3 v ( t ) 2 sin 2 (3.69) whic h is the dip ole radiation pattern. If the result of ( 3.65 ) is correct, then it should reduce to the Larmor form ula in the limit of the w a v epac k et width b going to zero

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54 (p oin t c harge). Making use of the iden tit y ( x ) = lim ` + 1 ` p e ` 2 x 2 ; (3.70) where ` = 1 = p 2 b; the dieren tial p o w er in ( 3.65 ) b ecomes dP d n = q 2 4 c 3 sin 2 h v f ( t t 0 ) ( t t 1 ) g i 2 | {z } a 2 : (3.71) Again v =c 1 w as assumed. The term in square brac k ets has the dimensions of acceleration. And so, ( 3.71 ) reduces to the Larmor result ( 3.69 ) for the step wise v elo cit y v ( t ) = v ( t t 0 )( t 1 t ) : These results are presen ted in [ 88 ].

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CHAPTER 4 CANONICAL STR UCTURE The go v erning equation of quan tum mec hanics is the Sc hr odinger equation [ 7 8 ]. In the minimal coupling prescription it is i = [ i r q A =c ] 2 2 m + V + q : ( 1.1 ) The dynamics of the scalar p oten tial and v ector p oten tial A are not describ ed b y this linear equation. Sp ecication of these p oten tials as w ell as the initial v alues for the w a v efunction casts the Sc hr odinger equation in to a w ell-dened b oundary v alue problem that is also a w ell-dened initial v alue problem. The go v erning equations of electro dynamics are Maxw ell's equations [ 9 ]: r E = 4 r B = 4 c J + E c r B = 0 r E + B c = 0 : ( 1.2 ) The dynamics of the c harge densit y and curren t densit y J are not describ ed b y these linear equations. Sp ecication of the external sources as w ell as the initial v alues for the electric and magnetic elds E and B satisfying r E = 4 and r B = 0 casts the Maxw ell equations in to a w ell-dened b oundary v alue problem that is also a w ell-dened initial v alue problem. Eac h of these theories are signican t in and of themselv es. Giv en a particular arrangemen t of sources throughout space-time and the initial v alues for E and B ; the Maxw ell equations go v ern the dynamics of the resulting electromagnetic eld. Lik ewise, giv en a particular external eld throughout space-time and the initial v alue for ; the Sc hr odinger equation go v erns the dynamics of the sources. Ho w ev er, notice that the Maxw ell equations do not sa y an ything ab out the dynamics of the sources and the Sc hr odinger equation do es not sa y an ything ab out the electro dynamics. 55

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56 It is p ossible to couple the linear Maxw ell and Sc hr odinger equations. The resulting nonlinear Maxw ell-Sc hr odinger theory accoun ts for the dynamics of the c harges and the electromagnetic eld as w ell as their m utual in teraction. F or example, giv en an initial source and its corresp onding Coulom b eld, a w a v efunction and electromagnetic eld are generated. The electromagnetic eld has its o wn dynamics and acts bac k up on the w a v efunction. This in turn causes dieren t elds to b e generated. It will b e demonstrated that these coupled nonlinear Maxw ell-Sc hr odinger equations can b e cast in to a w ell-dened initial v alue problem and solv ed in an ecien t n umerical manner. 4.1 Lagrangian Electro dynamics Consider the Maxw ell Lagrangian densit y L Max = [ A =c r ] 2 [ r A ] 2 8 + J A c (4.1) with external sources and J : V ariation of this Lagrangian leads to the go v erning equations of electro dynamics, i.e. r 2 A A c 2 r h r A + c i = 4 c J r 2 + r A c = 4 ( 1.4 ) These Maxw ell equations (in terms of the p oten tials) do not form a w ell-dened initial v alue problem. But, b y c ho osing a particular gauge they can b e turned in to one. In other w ords, these equations are ill-p osed as they stand. Ho w ev er, they do enjo y b oth Loren tz and gauge in v ariance as do es the Lagrangian ( 4.1 ). 4.1.1 Cho osing a Gauge W orking in a particular gauge can b e organized in to the follo wing hierarc h y: 1. A t the solution lev el, a gauge generator F can b e c hosen so that a gauge transformation of the solutions, i.e. 0 = F =c and A A 0 = A + r F ; maps them to new solutions that satisfy the gauge condition.

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57 2. A t the equation lev el, the set consisting of ( 1.4 ) together with a gauge constrain t has only solutions that satisfy the gauge condition. 3. A t the Lagrangian lev el, a gauge xing term can b e added to ( 4.1 ) so that the resulting Euler-Lagrange equations automatically include the gauge constrain t. 4.1.2 The Lorenz and Coulom b Gauges The rst t w o tiers can b e elab orated on as follo ws. With a gauge function F satisfying r 2 F F =c 2 = [ =c + r A ] a solution A = ( ; A ) of the p oten tial equations ( 1.4 ) can b e mapp ed to the Lorenz gauge solution A Lorenz according to the gauge transformation: Lorenz = F =c A A Lorenz = A + r F : (4.2) Alternativ ely adding the gauge constrain t =c + r A = 0 to ( 1.4 ) leads to the Lorenz gauge equations of motion: r 2 A A c 2 = 4 c J r 2 c 2 = 4 : (4.3) With and J sp ecied throughout space-time, the Lorenz gauge equations of motion are w ell-dened once the initial v alues for A ; A ; ; and are kno wn. There is some symmetry left in the solutions to these equations. Namely the residual gauge freedom left in the homogeneous equation r 2 F F =c 2 = 0 allo ws for gauge transformations on the solutions suc h that the new solutions do not lea v e the Lorenz gauge. Ho w ev er, these gauge transformed solutions do corresp ond to dieren t initial conditions. Note that the Lorenz gauge enjo ys relativistic or Loren tz in v ariance. It will b e sho wn, that the Lorenz gauge is the most appropriate gauge for dynamics. With another gauge function G satisfying r 2 G = r A a solution A = ( ; A ) of the p oten tial equations ( 1.4 ) can b e mapp ed to the Coulom b gauge solution A Coulom b according to: Coulom b = G=c A A Coulom b = A + r G: (4.4)

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58 Alternativ ely adding the gauge constrain t r A = 0 to ( 1.4 ) leads to the Coulom b gauge equations of motion: r 2 A A c 2 = 4 c J + r c r 2 = 4 : (4.5) Again with and J sp ecied throughout space-time, the Coulom b gauge equations of motion are w ell-dened once the initial v alues for A ; A ; ; and are kno wn. As b efore, there remains a symmetry or residual gauge freedom from the homogeneous equation r 2 G = 0 : Note that in the Coulom b gauge Gauss's la w reduces to r 2 = 4 : In v erting this equation sp ecies in terms of : That is = (1 = r 2 )[ 4 ] : The scalar p oten tial can no w b e totally remo v ed from the theory b y substitution of this Green's function in tegral. This ma y b e done at the exp ense of Loren tz in v ariance. In practice, where the equations are to b e expanded in a basis of s gaussians, either transv erse basis functions w ould ha v e to b e used or the transv erse elds w ould ha v e to b e generated from a standard basis. The former case w ould require a ma jor revision of most existing in tegral co des, whic h are in direct space, while the latter w ould require the instan taneous transv erse pro jection P ab T = ab @ a @ b = r 2 (see App endix B ) This op eration, whic h is o v er all space, is dicult to describ e in terms of a lo cal set of basis functions. Lastly for the third tier, consider the Lagrangian densit y ( 4.1 ) together with a gauge xing term for the Lorenz gauge, i.e. L LMax = L Max [ =c + r A ] 2 8 = [ A =c r ] 2 [ r A ] 2 8 + J A c [ =c + r A ] 2 8 : ( 1.18 ) The resulting Euler-Lagrange equations obtained from L LMax are iden tical to the Lorenz gauge w a v e equations in ( 4.3 ) whic h are equiv alen t to the general p oten tial equations ( 1.4 ) together with the constrain t =c + r A = 0 :

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59 PSfrag replacemen ts L Max L LMax L CMax gauge in v arian t d dt @ L Max @ @ L Max @ = 0 d dt @ L Max @ @ L Max @ = 0 d dt @ L Max @ @ L Max @ = 0 Maxw ell's equations w ell-p osed IVP w ell-p osed IVP w ell-p osed IVP unique unique solution solution solutions many r A = 0 r A + =c = 0 il l-p ose d IVP add add gauge gauge transformation transformation constrain t constrain t Figure 4{1: A limited but relev an t p ortion of the gauge story in the Lagrangian formalism is organized in this picture. The middle column ( i.e. the column b elo w L Max ) enjo ys full gauge freedom. The far left (Coulom b gauge) and far righ t (Lorenz gauge) columns ha v e limited gauge freedom. That is, there are a limited class of gauge transformations that can b e made on the solutions suc h that they remain in the same gauge. This symmetry is due to the residual gauge freedom. Note that these solutions corresp ond to dieren t initial conditions within the gauge. Also note that the Euler-Lagrange equations together with a particular gauge constrain t are equiv alen t to the Euler-Lagrange equations deriv ed from that particular gauge xed Lagrangian. There are man y other kno wn gauges, the c hoice of whic h is arbitrary All c hoices of gauge lead to the same ph ysically observ able electromagnetic elds E and B : T ogether with the denitions E = A =c r and B = r A ; the Lorenz and Coulom b gauge equations of motion as w ell as the general p oten tial equations ( 1.4 ) imply Maxw ell's equations ( 1.2 ). A diagram of this gauge story in the Lagrange form ulation is presen ted in Figure 4{1 4.2 Hamiltonian Electro dynamics In the Hamiltonian prescription, the momen tum conjugate to A with resp ect to the Maxw ell Lagrangian ( 4.1 ) is @ L Max @ A = 1 4 c [ A =c + r ] : (4.6)

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60 The momen tum conjugate to is iden tically zero, i.e. @ L Max @ = 0 : (4.7) A Hamiltonian densit y can still b e dened as the time-time comp onen t of the Maxw ell stress-energy tensor T Max = f @ L Max =@ ( @ ) g @ g L Max It is H Max T 00 Max = A + L Max = [ 4 c ] 2 + [ r A ] 2 8 c r + J A c (4.8) and the resulting equations of motion are: A @ H Max @ = 4 c 2 c r @ H Max @ A = r [ r A ] r 2 A 4 J c + c r (4.9) @ H Max @ = 0 @ H Max @ = + c r : Since the momen tum dened in ( 4.7 ) is iden tically zero, so is its time deriv ativ e and gradien t r : Notice that these Hamilton equations form a w ell-p osed initial v alue problem. The mac hinery inheren t in the Hamiltonian formalism automatically adds a momen tum and automatically adds the additional equation of constrain t = 0 : It turns out that this extra equation xes a particular gauge where = 0 : This gauge can alw a ys b e xed b y a gauge transformation whose generator satises F =c = : The residual gauge freedom left in the homogeneous equation F = 0 do es allo w for a gauge transformation on the solutions to ( 4.9 ). These new gauge transformed solutions do not lea v e the = 0 gauge, but do corresp ond to a dieren t initial v alue problem within this gauge. In other w ords, they are solutions to ( 4.9 ) with dieren t initial v alues. P a y careful atten tion to the fact that these Hamilton equations of motion form a w ell-p osed initial v alue problem ev en though a gauge xed Lagrangian w as not kno wingly used. The Hamiltonian formalism automatically added the extra equation = 0 :

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61 4.2.1 Hamiltonian F orm ulation of the Lorenz Gauge Rather than xing the Coulom b gauge at the equation lev el it ma y b e b enecial to w ork in a more general theory where a gauge is c hosen at the Lagrangian lev el and retains all of the 4-p oten tial, is Loren tz in v arian t, and do es not require an y instan taneous or nonlo cal op erations. T o this end, consider the Lorenz gauge Lagrangian densit y from ( 1.18 ), i.e. L LMax = [ A =c r ] 2 [ r A ] 2 8 + J A c [ =c + r A ] 2 8 : ( 1.18 ) It will b e sho wn that the equations of motion deriv ed from L LMax are w ell-dened b ecause of the addition of the last term in this expression. It turns out that this term is kno wn in the literature [ 24 68 ] and is a gauge xing term for the Lorenz gauge. F rom ( 1.18 ), the momen tum conjugate to A is @ L LMax @ A = 1 4 c [ A =c + r ] (4.10) and the momen tum conjugate to is @ L LMax @ = 1 4 c [ =c + r A ] : (4.11) With these momen ta and co ordinates, electro dynamics is giv en a symplectic structure. The Hamiltonian densit y is H L Max = [ 4 c ] 2 + [ r A ] 2 [4 c ] 2 8 c r c r A + J A c (4.12) and the resulting equations of motion are: A = 4 c 2 c r = r [ r A ] r 2 A 4 J c + c r (4.13) = 4 c 2 c r A = + c r :

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62 These equations, whic h are a generalization of ( 4.3 ), together with the initial v alues for A ; ; ; and form a w ell-p osed initial v alue problem. The residual gauge freedom resulting from the homogeneous equation F = 0 do es allo w for a gauge transformation on the solutions to ( 4.13 ). These new gauge transformed solutions do not lea v e the Lorenz gauge, but do corresp ond to a dieren t initial v alue problem within the Lorenz gauge. In other w ords, they are solutions to ( 4.13 ) with dieren t initial v alues. Notice that a relationship exists b et w een the momen tum and the gauge function F leading to the Lorenz gauge. That is, from = [ =c + r A ] = 4 c and F =c 2 r 2 F = =c + r A notice that F = 4 c: So the D'Alem b ertian of the gauge function F acts a generalized co ordinate in this phase space. It is the momen tum conjugate to the scalar p oten tial : In matrix form, the dynamical equations in ( 4.13 ) are 0BBBBBBB@ 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1CCCCCCCA 0BBBBBBB@ A _ 1CCCCCCCA = 0BBBBBBB@ r [ r A ] = 4 J =c + c r + c r 4 c 2 c r 4 c 2 c r A 1CCCCCCCA ; (4.14) where 1 is the 3 3 iden tit y matrix. Notice that ( 4.14 ) is of the Hamiltonian form = @ H =@ : (4.15) More sp ecically ab b = @ H =@ a ; where b is a column matrix of the generalized p ositions and momen ta, i.e. b ( x ; t ) = 0BBBBBBB@ A k k 1CCCCCCCA ; (4.16)

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63 where k = 1 ; 2 ; 3 : The an tisymmetric matrix ab is the (canonical) symplectic form asso ciated with the phase space of electro dynamics in the Lorenz gauge. By substitution, these rst order Hamiltonian equations of motion can b e sho wn to b e equivalen t to the second order Lorenz gauge equations = 4 and A = 4 J =c: T ogether with the denition of the electric and magnetic elds, ( 4.13 ) imply r E = 4 + 4 r B = 4 c J + E c 4 c r (4.17) r B = 0 r E + B c = 0 : These equations are not equiv alen t to Maxw ell's equations unless ( x ; t ) remains constan t in space-time throughout the dynamics. In order to analyze this question, the dynamics of the sources m ust b e considered. It should b e noticed that the inhomogeneous equations in ( 4.17 ) imply r 2 c 2 = 1 c 2 [ + r J ] : (4.18) If the matter theory is suc h that the equation of con tin uit y = r J is satised, then = 0 : So if ( t = 0) = ( t = 0) = 0 ; then ( t ) = 0 at all times t: In other w ords, if the sources of c harge and curren t satisfy the equation of con tin uit y then the dynamical theory arising from the Lagrangian ( 1.18 ) is the Maxw ell theory of electro dynamics. Note that while ( 4.9 ) and ( 4.13 ) do not enjo y the full gauge symmetry as do the general p oten tial equations ( 1.4 ), this do es not mean that the observ ables resulting from ( 4.9 ) or ( 4.13 ) are not gauge in v arian t. An y observ able that is calculated will b e in v arian t to the c hoice of gauge generator. Moreo v er, once the solutions to these w ell-dened equations are constructed, these solutions b elong to the man y solutions of ( 1.4 ). This family of solutions is the most general solutions of the p oten tial form of Maxw ell's equations. In fact, gauge transformations can ev en b e made from one

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64 particular gauge to another [ 89 ]. A diagram depicting the relev an t gauge story in the Hamiltonian form ulation is presen ted in Figure 4{2 Notice that there is no Hamiltonian theory that enjo ys the full gauge symmetry of ( 1.4 ). The Hamiltonian H Max in the far righ t column is obtained b y a Legendre transformation of the gauge in v arian t Lagrangian L Max in ( 4.1 ). Ho w ev er, the Hamiltonian dynamics stemming from the gauge in v arian t L Max is not gauge in v arian t, but rather o ccurs in the gauge where = 0 :

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65 PSfrag replacemen ts LMaxLLMaxLCMaxgauge in v arian td dt @ LMax @ @ LMax @ = 0d dt @ LMax @ @ LMax @ = 0d dt @ LMax @ @ LMax @ = 0 Maxw ell's equations w ell-p osed IVP w ell-p osed IVP w ell-p osed IVP w ell-p osed IVP w ell-p osed IVP w ell-p osed IVP unique unique unique solution solution solution solutions many r A = 0 r A + =c = 0 il l-p ose d IVP add add constrain t constrain t gauge gauge gauge transformation transformation transformation =@ HMax @ =@ HLMax @ =@ HCMax @ HMax=@ LMax @ _ LMaxHL Max=@ LLMax @ _ LLMaxHC Max=@ LCMax @ _ LCMaxFigure 4{2: The Hamiltonian form ulation of the gauge story is organized in this picture with resp ect to the previous Lagrangian form ulation. Figure 4{1 is depicted in the b o x with dotted b orders. It can no w b e seen ho w the Coulom b and Lorenz gauges connect in b oth formalisms.

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66 4.2.2 P oisson Brac k et for Electro dynamics The phase space that carries the asso ciated dynamics is naturally endo w ed with a P oisson brac k et f ; g (recall Chapter 2 ). This ma y b e seen b y considering the v ariation of along the dynamics H ( @ =@ ) That is H ( ) ( d=dt ) = ( @ =@ b ) b = ( @ =@ b ) 1 ab ( @ H =@ a ) f ; H g ; (4.19) where are the generalized co ordinates. In general, the P oisson brac k et of the dynamical v ariable F with the dynamical v ariable G is f F ; G g = 0BBBBBBB@ @ F =@ A @ F =@ @ F =@ @ F =@ 1CCCCCCCA T 0BBBBBBB@ 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1CCCCCCCA 1 0BBBBBBB@ @ G=@ A @ G=@ @ G=@ @ G=@ 1CCCCCCCA : (4.20) Since the symplectic form is canonical its in v erse is trivial, i.e. 1 = T = : Also notice that 2 = 1 ; T = 1 ; and det = 1 : 4.3 Hamiltonian Electro dynamics and W a v e Mec hanics in Complex Phase Space Consider the matter theory asso ciated with the Sc hr odinger Lagrangian ( ~ = 1) L Sc h = i [ i r q A =c ] [ i r q A =c ] 2 m V q ( 1.17 ) where is the w a v efunction for a single electron, V = q q = j x j is the static Coulom b p oten tial energy of a proton, and ( ; A ) are the electron's scalar and v ector p otentials. Notice that this Lagrangian is already written in phase space. The momen tum conjugate to the w a v efunction is i : T ogether with the previous Maxw ell Lagrangian, the coupled nonlinear dynmical theory arising from the Lagrangians L Max = 1 2 [ A A ] n [ 4 c ] 2 + [ r A ] 2 8 c r o (4.21)

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67 L Sc h = i 2 [ _ ] n [ i r q A =c ] [ i r q A =c ] 2 m + V + q o (4.22) L gauge = 1 2 [ _ ] f 2 c 2 2 c r A g (4.23) yields the follo wing equations of motion: A = 4 c 2 c r = r [ r A ] r 2 A 4 J c + c r = 4 c 2 c r A = + c r (4.24) i = [ i r q A =c ] 2 2 m + V + q i = [ i r q A =c ] 2 2 m + V + q : Surface terms of the form ( d=dt ) f pq = 2 g ha v e b een added in the ab o v e Lagrangians in order to symmetrize them, i.e. L = p q H ( d=dt ) f pq = 2 g b ecomes L = [ p q pq ] = 2 H : This can alw a ys b e done since the action I = R Ldt = R [ L + ( d=dt ) g ] dt is in v arian t to the addition of a pure surface term to the Lagrangian. Note that the Sc hr odinger w a v efunctions and are complex-v alued while the remaining electromagnetic v ariables are all real-v alued. These dynamical equations ma y b e put in to matrix form as 0BBBBBBBBBBBBBB@ i 0 0 0 0 0 0 i 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1CCCCCCCCCCCCCCA 0BBBBBBBBBBBBBB@ _ A _ 1CCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBB@ [ i r q A =c ] 2 = 2 m + V + q [ i r q A =c ] 2 = 2 m + V + q r [ r A ] = 4 J =c + c r + c r 4 c 2 c r 4 c 2 c r A 1CCCCCCCCCCCCCCA ; (4.25) where the symplectic form is canonical. The electromagnetic sector of it is iden tical to ( 4.14 ). These dynamical equations dene the coupled Maxw ell-Sc hr odinger

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68 theory This theory is w ell-dened and closed. In other w ords, the dynamics of the c harges, curren ts, and elds are all sp ecied as w ell as their m utual in teraction. Giv en initial v alues for ; ; A ; ; ; and determines their coupled dynamics throughout space-time. With the dynamics of the c harges dened, the problem in ( 4.17 ) can no w b e addressed. The Sc hr odinger equation in ( 4.24 ) implies the con tin uit y equation ( d=dt ) q = r q [ i r q A =c ] + [ i r q A =c ] = 2 m (4.26) whic h ma y b e written more compactly as = r J : F rom the denition of the momen tum in ( 4.11 ) and the w a v e equations = 4 and A = 4 J =c; notice that = 1 4 c [( d=dt ) =c + r A ] = 1 4 c [( d=dt )4 =c + r 4 J =c ] = 1 c 2 [ + r J ] = 0 (4.27) b y app ealing to ( 4.26 ). So if ( t = 0) = ( t = 0) = 0 ; then the electro dynamics sta ys in the Lorenz gauge for all time since the only solution of = 0 with ( t = 0) = ( t = 0) = 0 is ( t ) = 0 : It is w orth men tioning that if ( t = 0) = 0 for all time, then the electronelectron self in teraction mak es no con tribution to the Sc hr odinger energy This is true since the self in teraction term q in the ab o v e Sc hr odinger Lagrangian cancels exactly with c r in the Maxw ell Lagrangian. The cancellation requires a partial in tegration of c r to c r follo w ed b y a substitution of 0 = + c r from ( t = 0) = 0 in ( 4.24 ). Ho w ev er, there is still a con tribution from the self-energy arising in the Maxw ell energy of the Coulom bic eld.

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69 4.4 Hamiltonian Electro dynamics and W a v e Mec hanics in Real Phase Space The dynamical equations ( 5.16 ) are mixed, real and complex. F or consistency these equations are put in to real form with the Lagrangian densities: L Max = 1 2 [ A A ] n [ 4 c ] 2 + [ r A ] 2 8 c r o (4.28) L Sc h = 1 2 [ P Q P Q ] 8><>: f [ r Q + q A P =c ] 2 + [ r P + q A Q=c ] 2 g = 4 m + V [ Q 2 + P 2 ] = 2 + q [ Q 2 + P 2 ] = 2 9>=>; (4.29) L gauge = 1 2 [ _ ] f 2 c 2 2 c r A g (4.30) The functions P and Q are related to the real and imaginary parts of and according to = [ Q + iP ] = p 2 and = [ Q iP ] = p 2 : The equations of motion that are asso ciated with these Lagrangians are: A = 4 c 2 c r = r [ r A ] r 2 A 4 J c + c r (4.31a) = 4 c 2 c r A = + c r Q = r 2 P + q r ( A Q ) =c + q A r Q=c + q 2 A 2 P =c 2 2 m + V P + q P (4.31b) P = r 2 Q + q r ( A P ) =c + q r P A =c q 2 A 2 Q=c 2 2 m + V Q + q Q:

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70 These dynamical equations ma y b e put in to matrix form as 0BBBBBBBBBBBBBB@ 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1CCCCCCCCCCCCCCA 0BBBBBBBBBBBBBB@ A _ Q _ P 1CCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBB@ r [ r A ] = 4 J =c + c r + c r [ r 2 Q + q r ( A P ) =c + q r P A =c q 2 A 2 Q=c 2 ] = 2 m + V Q + q Q 4 c 2 c r 4 c 2 c r A [ r 2 P + q r ( A Q ) =c + q A r Q=c + q 2 A 2 P =c 2 ] = 2 m + V P + q P 1CCCCCCCCCCCCCCA ; (4.32) where the symplectic form is again canonical. Note that the equation of con tin uit y = r J still holds with the real c harge and curren t densities = q [ Q 2 + P 2 ] = 2 J = q 2 m f Q r P P r Q q Q A Q=c q P A P =c g : (4.33) 4.5 The Coulom b Reference b y Canonical T ransformation As w as men tioned previously the n umerical implemen tation of the theory can b e made to con v erge more quic kly if the basis is c hosen judiciously Recall that the electromagnetic eld generated b y an y c harge con tains a Coulom bic con tribution. This monop ole term accoun ts for a large p ortion of the lo cal electromagnetic eld surrounding the c harge. It w ould b e adv an tageous to not describ e this large

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71 con tribution in terms of the basis but rather to calculate it analytically The remaining smaller p ortion of the radiativ e or dynamical electromagnetic eld can then b e describ ed in terms of the basis. T o this end, notice that the scalar p oten tial = C + ( C ) C + D ma y b e split in to a Coulom bic p ortion satisfying r 2 C = 4 that can b e calculated analytically and a remainder D regardless of the c hoice of gauge. The Coulom bic p oten tial is not itself a dynamical v ariable but dep ends on the dynamical v ariables Q and P : That is C ( x ; t ) = R V d 3 x 0 j x x 0 j 1 q [ Q ( x 0 ; t ) Q ( x 0 ; t ) + P ( x 0 ; t ) P ( x 0 ; t )] = 2 : The dynamical p ortion D is a generalized co ordinate and is represen ted in the basis. Similarly the momen tum conjugate to A ma y b e split in to a Coulom bic and dynamical piece according to @ L Max @ A = C + D = r C 4 c + 1 4 c [ A =c + r D ] : (4.34) Lik e D ; the dynamical p ortion D is a generalized co ordinate and is represen ted in the basis. 4.5.1 Symplectic T ransformation to the Coulom b Reference The transformation to these new co ordinates, i.e. D and D is obtained b y the canonical or symplectic transformation T : 0BBBBBBBBBBBBBB@ A Q P 1CCCCCCCCCCCCCCA 0BBBBBBBBBBBBBB@ ~ A ( A ) ~ ( ; Q; P ) ~ Q ( Q ) ~ ( ; Q; P ) ~ () ~ P ( P ) 1CCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBB@ A C ( Q; P ) Q C ( Q; P ) P 1CCCCCCCCCCCCCCA ; (4.35) where ~ D and ~ D : The v ariables Q; P ; A ; and are unc hanged b y T : Since b oth C and C are complicated functions of Q and P ; the in v ersion of T ma y

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72 b e quite in v olv ed. Ho w ev er, it will b e sho wn that the in v erse of T do es exist. In fact b oth the T and T 1 are dieren tiable mappings on symplectic manifolds. Therefore the canonical transformation is a symplectic dieomorphism or symplectomorphism [ 29 ]. The theory of restricted ( i.e. explicitly time-indep enden t) canonical transformations [ 27 28 ] giv es the general prescription for the transformation of the old Hamilton equations ( 4.32 ) to the new Hamilton equations in terms of T (and T T ) only In sym b ols, that is = 1 @ H @ ~ = ~ 1 @ ~ H @ ~ ; (4.36) where the new Hamiltonian ~ H is equiv alen t to the old Hamiltonian H expressed in terms of the new v ariables ~ : (F or simplicit y ~ H will b e written as H from this p oin t forw ard.) T o this end, consider the time deriv ativ e of the new column matrix ~ i = @ ~ i @ j j T ij j or ~ = T : (4.37) Substituting from ( 4.36 ) results in ~ i = T ij 1 j k @ H @ k or ~ = T 1 @ H @ : (4.38) Lastly the column matrix @ H =@ can b e written as @ H @ k = @ H @ ~ l @ ~ l @ k T T k l @ H @ ~ l or @ H @ = T T @ H @ ~ (4.39) so that the new equations of motion ( 4.38 ) b ecome ~ i = T ij 1 j k T T k l @ H @ ~ l or ~ = T 1 T T @ H @ ~ ~ 1 @ H @ ~ : (4.40)

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73 This canonical transformation on the equations of motion lea v es only the computation of ~ 1 T 1 T T since the Hamiltonian automatically b ecomes H = [ 4 c f ~ + C ( ~ Q ; ~ P ) g ] 2 + [ r ~ A ] 2 [4 c ~ ] 2 8 + q [ ~ + C ( ~ Q ; ~ P )] ~ Q 2 + ~ P 2 2 c r [ ~ + C ( ~ Q; ~ P )] [ ~ + C ( ~ Q; ~ P )] c ~ r ~ A + [ r ~ Q + q ~ A ~ P =c ] 2 + [ r ~ P + q ~ A ~ Q =c ] 2 4 m + V ~ Q 2 + ~ P 2 2 (4.41) in terms of the new co ordinates. Ho w ev er, the in v ersion of is not simple in practice. It turns out that the equations of motion ( 4.36 ) are most practically written as @ H @ ~ = ~ ~ = ( T 1 ) T T 1 ~ not ~ = ~ 1 @ H @ ~ = T 1 T T @ H @ ~ ; (4.42) where the in v erse transformation T 1 is the transformation of the in v erse mapping, i.e. T 1 ij @ i =@ ~ j : It will b e sho wn that det T 6 = 0 so the mapping is w ell-dened. These equations of motion are of the desired form b ecause they in v olv e and not 1 : That 1 is undesirable is seen b y going to the basis. In the basis, the canonical symplectic form b ecomes 0 1 1 0 0B@ 0 @ 2 h Q j P i @ q K @ p J @ 2 h P j Q i @ p K @ q J 0 1CA (4.43) whic h is not easily in v erted. As a result it is simpler to compute ( T 1 ) T T 1 than T 1 T T ev en though T 1 is needed in the former case. It will b e sho wn that the explicit ev aluation of T 1 is not necessary T o con tin ue with the transformed equations of motion in ( 4.42 ), whic h only require ; the mapping ( T 1 ) T : @ =@ @ =@ ~ m ust rst b e set up. The transp osed in v erse transformation ( T 1 ) T is dened on the v ector elds themselv es according

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74 to 0BBBBBBBBBBBBBB@ @ =@ ~ A @ =@ ~ @ =@ ~ Q @ =@ ~ @ =@ ~ @ =@ ~ P 1CCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBB@ @ A =@ ~ A 0 0 0 0 0 0 @ =@ ~ 0 0 0 0 0 @ =@ ~ Q @ Q=@ ~ Q @ =@ ~ Q 0 0 0 0 0 @ =@ ~ 0 0 0 0 0 0 @ =@ ~ 0 0 @ =@ ~ P 0 @ =@ ~ P 0 @ P =@ ~ P 1CCCCCCCCCCCCCCA 0BBBBBBBBBBBBBB@ @ =@ A @ =@ @ =@ Q @ =@ @ =@ @ =@ P 1CCCCCCCCCCCCCCA : (4.44) Notice that det ( T 1 ) T = det T 1 = (det T ) 1 @ ( A ; ; Q; ; ; P ) @ ( ~ A ; ~ ; ~ Q; ~ ; ~ ; ~ P ) = ( @ ~ A =@ A )( @ ~ =@ )( @ ~ Q =@ Q )( @ ~ =@ )( @ ~ =@ )( @ ~ P =@ P ) = 1 (4.45) so that the transformation is canonical and symplectic or area preserving. In other w ords, the new innitesimal v olume elemen t d ~ is related to the old innitesimal v olume elemen t d b y d ~ = det T d = d (4.46) since the determinan t of the Jacobian is unit y Th us, the v olume elemen t of phase space is the same b efore and after the transformation. It is a canonical in v arian t.

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75 With ( T 1 ) T the similarit y transformation of the canonical symplectic form in ( 4.32 ) is ~ ( T 1 ) T T 1 = n M n > n n G ; (4.47) where n > = 0BBBB@ ( @ A =@ ~ A )( 1 )( @ =@ ~ ) 0 ( @ A =@ ~ A )( @ =@ ~ P ) 0 ( @ =@ ~ )( 1)( @ =@ ~ ) 0 0 ( @ =@ ~ Q )( @ =@ ~ ) ( @ Q =@ ~ Q )( 1)( @ P =@ ~ P ) 1CCCCA (4.48) n = 0BBBB@ ( @ =@ ~ )( 1 )( @ A =@ ~ A ) 0 0 0 ( @ =@ ~ )(1)( @ =@ ~ ) ( @ =@ ~ )( @ =@ ~ Q ) ( @ =@ ~ P )( @ A =@ ~ A ) 0 ( @ P =@ ~ P )(1)( @ Q =@ ~ Q ) 1CCCCA (4.49) n M = 0BBBB@ 0 0 ( @ A =@ ~ A )( @ =@ ~ Q ) 0 0 0 ( @ =@ ~ Q )( @ A =@ ~ A ) 0 0 1CCCCA (4.50) n G = 0BBBB@ 0 0 0 0 0 ( @ =@ ~ )( @ =@ ~ P ) 0 ( @ =@ ~ P )( @ =@ ~ ) 0 1CCCCA : (4.51) The factors of 1 and 1 are explicitly written in n > and n to bring out their similarit y to the canonical symplectic form in ( 4.32 ). After computing the deriv ates

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76 in ~ it can b e sho wn that ~ equals 0BBBBBBBBBBBBBB@ 0 0 @ C ( ~ Q; ~ P ) =@ ~ Q 1 0 @ C ( ~ Q; ~ P ) =@ ~ P 0 0 0 0 1 0 @ C ( ~ Q; ~ P ) =@ ~ Q 0 0 0 @ C ( ~ Q; ~ P ) =@ ~ Q 1 1 0 0 0 0 0 0 1 @ C ( ~ Q ; ~ P ) =@ ~ Q 0 0 @ C ( ~ Q; ~ P ) =@ ~ P @ C ( ~ Q; ~ P ) =@ ~ P 0 1 0 @ C ( ~ Q; ~ P ) =@ ~ P 0 1CCCCCCCCCCCCCCA (4.52) with C ( x ; t ) = q 2 R V Q ( x 0 ; t ) 2 + P ( x 0 ; t ) 2 j x x 0 j d 3 x 0 (4.53) and C ( x ; t ) = q 8 c r R V Q ( x 0 ; t ) 2 + P ( x 0 ; t ) 2 j x x 0 j d 3 x 0 : (4.54) And so the new symplectic form con tains extra elemen ts that are not presen t in the canonical : These extra elemen ts add additional time-dep enden t couplings to the theory As b efore, the asso ciated phase space is naturally endo w ed with the P oisson brac k et f F ; G g ~= ( @ F =@ ~ ) T ~ 1 ( @ G=@ ~ ) : (4.55)

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77 The transformed equations of motion with symplectic form ( 4.52 ) ma y b e written in full as: @ C @ ~ Q ~ Q ~ @ C @ ~ P ~ P = @ H @ ~ A = r [ r ~ A ] 4 + c r ~ q 2 mc n ~ Q r ~ P ~ P r ~ Q q c ~ Q ~ A ~ Q q c ~ P ~ A ~ P o ~ = @ H @ ~ = q ~ Q 2 + ~ P 2 2 + c r [ ~ + C ] @ C @ ~ Q ~ A @ C @ ~ Q ~ ~ P = @ H @ ~ Q = r 2 ~ Q q r ( ~ A ~ P ) =c q r ~ P ~ A =c + q 2 ~ A 2 ~ Q=c 2 2 m + V ~ Q + q [ ~ + C ] ~ Q + f 4 c 2 [ ~ + C ] c r [ ~ + C ] g @ C @ ~ Q + n q ~ Q 2 + ~ P 2 2 + c r [ ~ + C ] o @ C @ ~ Q ~ A = @ H @ ~ = 4 c 2 [ ~ + C ] c r [ ~ + C ] ~ + @ C @ ~ Q ~ Q + @ C @ ~ P ~ P = @ H @ ~ = 4 c 2 ~ c r ~ A @ C @ ~ P ~ A + ~ Q @ C @ ~ P ~ = @ H @ ~ P = r 2 ~ P + q r ( ~ A ~ Q ) =c + q ~ A r ~ Q=c + q 2 ~ A 2 ~ P =c 2 2 m + V ~ P + q [ ~ + C ] ~ P + f 4 c 2 [ ~ + C ] c r [ ~ + C ] g @ C @ ~ P + n q ~ Q 2 + ~ P 2 2 + c r [ ~ + C ] o @ C @ ~ P ; (4.56) where C C ( ~ Q; ~ P ) and C C ( ~ Q; ~ P ) : The forces app earing on the righ t hand side of these equations ha v e b ecome more complicated, esp ecially those in the Sc hr odinger equations. There are new nonlinear terms. Ho w ev er, it is p ossible to substitute these equations among themselv es in order to simplify them. Notice that parts of the @ H =@ ~ and @ H =@ ~ equations app ear in the forces of the Sc hr odinger

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78 equations. Substitution of @ H =@ ~ and @ H =@ ~ in to the Sc hr odinger equations results in the follo wing simplied equations: @ C @ ~ Q ~ Q ~ @ C @ ~ P ~ P = r [ r ~ A ] 4 + c r ~ q 2 mc n ~ Q r ~ P ~ P r ~ Q q c ~ Q ~ A ~ Q q c ~ P ~ A ~ P o ~ = q ~ Q 2 + ~ P 2 2 + c r [ ~ + C ] ~ P = r 2 ~ Q + q r ( ~ A ~ P ) =c + q r ~ P ~ A =c q 2 ~ A 2 ~ Q=c 2 2 m + V ~ Q + q [ ~ + C ] ~ Q ~ A = 4 c 2 [ ~ + C ] c r [ ~ + C ] ~ + @ C @ ~ Q ~ Q + @ C @ ~ P ~ P = 4 c 2 ~ c r ~ A ~ Q = r 2 ~ P + q r ( ~ A ~ Q ) =c + q ~ A r ~ Q =c + q 2 ~ A 2 ~ P =c 2 2 m + V ~ P + q [ ~ + C ] ~ P : (4.57) The generalized forces app earing on the righ t hand side are no w v ery similar to the forces in ( 4.31 ). In fact, the equations of motion ( 4.57 ) can b e further simplied as: [ + C ( Q; P )] = @ H =@ A A = @ H =@ = @ H =@ + C ( Q; P ) = @ H =@ P = @ H =@ Q Q = @ H =@ P ; (4.58) where the tildes w ere omitted to sho w the resem blance b et w een ( 4.58 ) and ( 4.31 ). 4.5.2 The Coulom b Reference b y Change of V ariable It can b e sho wn that the new equations of motion ~ ~ = @ H =@ ~ ; whic h w ere obtained b y a symplectic transformation in phase space, ma y also b e obtained b y a c hange of v ariable in the Lagrangians ( 4.28 )-( 4.30 ). The new Lagrangian densit y is:

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79 PSfrag replacemen ts L ( p; q ) ~ L ( ~ p; ~ q ) = @ H =@ ~ ~ = @ H =@ ~ Canonical T ransformation R [ p q H ( p; q )] dt = 0 R [ ~ p ~ q H ( ~ p; ~ q )] dt = 0Change of V ariables Figure 4{3: Comm utivit y diagram represen ting the c hange of co ordinates ( q ; p ) to ( ~ p; ~ q ) at b oth the Lagrangian and equation of motion lev els. ~ L Max = 1 2 [( ~ + C ) ~ A ( ~ + ~ C ) ~ A ] 8><>: f [ 4 c ( ~ + C )] 2 + [ r ~ A ] 2 g = 8 c r [ ~ + C ] [ ~ + C ] 9>=>; (4.59) ~ L Sc h = 1 2 [ ~ P ~ Q ~ P ~ Q ] 8><>: f [ r ~ Q + q ~ A ~ P =c ] 2 + [ r ~ P + q ~ A ~ Q=c ] 2 g = 4 m +[ V + q ( ~ + C )] f ~ Q 2 + ~ P 2 g = 2 9>=>; (4.60) ~ L gauge = 1 2 [ ~ ( ~ + C ) ~ ( ~ + C )] f 2 c 2 ~ 2 c ~ r ~ A g : (4.61) That the transformation to the Coulom b reference holds at b oth Lagrangian and equation of motion lev el demonstrates the comm utivit y of the diagram in Figure 4{3 4.6 Electron Spin in the P auli Theory The electron eld used so far in the nonrelativistic Sc hr odinger theory is a eld of spin zero, i.e. a scalar eld. It is a simple generalization of the theory to add in the electron's spin. The electron eld w ould then b e a t w o comp onen t spinor eld,

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80 i.e. a spin-1/2 eld, and w ould b e of the form P ( x ; t ) = ( x ; t ) # ( x ; t ) : (4.62) The rst comp onen t is spin up and the second comp onen t # is spin do wn. The dynamics of P is go v erned b y the P auli equation [ 30 ] i P = [ i r q A =c ] 2 P 2 m + V P + q P q 2 mc [ r A ] P (4.63) whic h is the nonrelativistic limit of the Dirac equation i D = mc 2 D + c [ i r q A =c ] D + q D (4.64) in terms of the four comp onen t spinor D where the and matrices are = I 0 0 I = 0 0 (4.65) and x = 0 1 1 0 y = 0 i i 0 z = 1 0 0 1 : (4.66) Notice that taking the nonrelativistic limit of the Dirac equation in v olv es the elimination of the t w o comp onen t p ositron eld from D : Also note that the curren t densit y asso ciated with the P auli theory [ 90 ] is dieren t from that in the Sc hr odinger theory (see ( 1.14 )). It is J P = q yP [ i r q A =c ] P + P [ i r q A =c ] yP + r [ yP P ] = 2 m; (4.67) where yP = ( # ) is the adjoin t of P : This can b e deriv ed b y taking the nonrelativistic limit of the Dirac curren t densit y The last term in ( 4.67 ) is only presen t in the P auli curren t. This term do es not eect the con tin uit y equation = r J since r r [ yP P ] = 0 :

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81 4.7 Proton Dynamics In the theory set up so far, the matter dynamics w as en tirely describ ed b y the electronic w a v efunction : The proton had no dynamics whatso ev er. Only the electrostatic scalar p oten tial q = q = j x j of the structureless proton of c harge q en tered so as to bind the electron in the h ydrogen atom. A rst step in the direction of atomic and molecular collisions requires the dynamics of the proton as w ell (and ev en tually a few other particles). Supp ose the proton is describ ed b y its o wn w a v efunction n and Lagrangian densit y L q Sc h = i n n [ i r q A =c ]n [ i r q A =c ]n 2 m q q n n ; (4.68) where ( ; A ) are the scalar and v ector p oten tials arising from the c harge and curren t densities = q + q n n (4.69) J = q [ i r q A =c ] + [ i r q A =c ] = 2 m q + q n [ i r q A =c ]n + n[ i r q A =c ]n = 2 m q : (4.70) These densities are just the sum of the individual electronic and proton densities. The proton densit y is not a delta function. Th us, the proton w a v efunction is not a delta function either. Rather it is describ ed b y a w a v epac k et and has some structure. With ( 4.68 ) the total Lagrangian is L Max = 1 2 [ A A ] n [ 4 c ] 2 + [ r A ] 2 8 c r o (4.71) L qSc h = i 2 [ _ ] n [ i r q A =c ] [ i r q A =c ] 2 m q + q o (4.72)

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82 L q Sc h = i 2 [n n n n] n [ i r q A =c ]n [ i r q A =c ]n 2 m q + q n n o (4.73) L gauge = 1 2 [ _ ] f 2 c 2 2 c r A g : (4.74) Notice that the electron Lagrangian ( 4.72 ) do es not explicitly con tain the static proton p oten tial energy V = q q = j x j as did the previous Sc hr odinger Lagrangian ( 1.17 ). The t w o matter elds are coupled en tirely through electro dynamics. That is, the electron-proton in teraction is mediated b y the electro dynamics. The Coulom bic p oten tial is included implicitly in q and q n n in the ab o v e matter Hamiltonians. In other w ords, the scalar p oten tial con tains (in an y gauge) a Coulom b piece of the form C ( x ; t ) = + ( C ) = R V ( x 0 ; t ) j x x 0 j d 3 x 0 = R V q ( x 0 ; t )( x 0 ; t ) j x x 0 j d 3 x 0 + R V q n ( x 0 ; t )n( x 0 ; t ) j x x 0 j d 3 x 0 : (4.75) With this p oten tial, the q term in the electron Hamiltonian con tains the electron-proton attraction as w ell as electron-electron self in teraction. Similarly the q n n in the proton Hamiltonian con tains the electron-proton attraction and proton-proton self in teraction. The self-energies that are computed from the aforemen tioned self in teractions are nite b ecause and n are square in tegrable functions. That is E in t = R V ( x ; t ) C ( x ; t ) d 3 x = R V d 3 x R V d 3 x 0 ( x ; t ) ( x 0 ; t ) j x x 0 j < 1 (4.76) for b oth the cross terms (electron-proton attraction) and the direct terms (electronelectron and proton-proton repulsion). Note that in the relativistic quan tum theory the direct terms are innite and there are innitely man y Coulom b states of the

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83 bare problem to sum o v er [ 40 ]. These innities do not arise in the semiclassical theory presen ted in this dissertation. While the self in teractions do app ear in the ab o v e matter Hamiltonians, the resulting self-energies are nite and moreo v er do not ev en con tribute to the electron or proton p ortions of the energy This is due to c r in the ab o v e Maxw ell Hamiltonian. After a partial in tegration this term b ecomes c r : Substitution of = + c r = 0 from ( 4.31 ) turns c r in to ; whic h cancels + in the electron and proton energies. Ho w ev er, the self in teractions do remain in the Coulom b energy E 2 = 8 of the electromagnetic eld. Note that the self in teractions do app ear in the Hamiltonians and therefore do mak e a con tribution to the o v erall dynamics. It should b e men tioned that this theory of electron-proton dynamics can b e applied to electron-p ositron dynamics as w ell. While there is a 2000-fold dierence in mass b et w een the proton and the p ositron, the t w o theories are otherwise iden tical. In either case, the theory ma y b e ric h enough to capture b ound states of h ydrogen or p ositronium.

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CHAPTER 5 NUMERICAL IMPLEMENT A TION The formal theory of Maxw ell-Sc hr odinger dynamics w as constructed in the previous c hapter. In particular, the coupled and nonlinear Maxw ell-Sc hr odinger equations i = [ P q A =c ] 2 2 m + V + q (5.1) r 2 A A c 2 r h r A + c i = 4 c J (5.2a) r 2 + r A c = 4 (5.2b) w ere recognized to b e ill-p osed unless an extra equation of constrain t is added to them. Using the Hamiltonain approac h to dynamics, this extra equation w as automatically generated b y adding a Lorenz gauge xing term at the Lagrangian lev el. It w as emphasized in Chapter 4 that the resulting Hamiltonian system of dieren tial equations, whic h are of rst order in time, form a w ell-dened initial v alue problem. That is, the Maxw ell-Sc hr odinger dynamics are kno wn in principle once the initial v alues are sp ecied for eac h of the dynamical v ariables. The details of con v erting the formal mathematics of Chapter 4 to a form that can b e practically implemen ted in a computer are presen ted in this c hapter. The Hamiltonian system of partial dieren tial equations will b e reduced to a Hamiltonian system of ordinary dieren tial equations in time b y in tro ducing a spatial basis for eac h of the dynamical v ariables. The resulting basis equations are co ded in a F or tran 90 computer program. With this program, v arious pictures are made to depict the dynamics of the h ydrogen atom in teracting with the electromagnetic eld. 84

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85 5.1 Maxw ell-Sc hr odinger Theory in a Complex Basis Eac h of the Maxw ell-Sc hr odinger dynamical v ariables, whic h are themselv es elds, ma y b e expanded in to a complete basis of functions G K according to ( x ; t ) = P K G K ( x ) K ( t ) ( x ; t ) = P K G K ( x ) K ( t ) A k ( x ; t ) = P K G K ( x ) a k K ( t ) k ( x ; t ) = P K G K ( x ) k K ( t ) ( x ; t ) = P K G K ( x ) K ( t ) ( x ; t ) = P K G K ( x ) K ( t ) ; (5.3) where the index K runs o v er the basis and the index k runs o v er 1, 2, 3 or x; y ; z : An y complete set of functions suc h as the oscillator eigenstates will suce. In the follo wing w ork the set of gaussian functions of the form G K ( x ) = G K ( x ) = N K exp ( ` K [ x r K ] 2 ) (5.4) are used. These functions are cen tered on r K ; normalized to unit y b y N K ; and are real-v alued. Additionally they span L 2 so that an y square in tegrable function ma y b e represen ted in this basis. In principle the sums in ( 5.3 ) are to innit y Ho w ev er, a complete basis cannot b e realized in practice. But for all practical purp oses the n umerical results can b e sho wn to con v erge to within an arbitrary accuracy in a nite basis. In fact with a smart c hoice of basis, the n umerical results ma y con v erge with just a few terms. Here the basis co ecien ts, whic h are complexand real-v alued as w ell as time-dep enden t, carry the dynamics. The basis represen tation of the previous Lagrangians is L Max = P KM 1 2 [( @ =@ a m M ) a m M ( @ =@ m M ) m M ] S Max H Max (5.5) L Sc h = P K i 2 [( @ =@ K ) K ( @ =@ K ) K ] S Sc h H Sc h (5.6) L gauge = P KM 1 2 [( @ =@ K ) K ( @ =@ K ) K ] S gauge H gauge (5.7)

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86 with in tegrals S Max = R V A d 3 x S Sc h = R V d 3 x S gauge = R V d 3 x: (5.8) The calculus of v ariations leads to the follo wing dynamical equations: @ 2 S Max @ m M @ a n N a n N = @ H @ m M or M m M ;n N a n N = r m M H (5.9) @ 2 S gauge @ I @ L L = @ H @ I or N I L L = r I H (5.10) @ 2 iS Sc h @ I @ L L = @ H @ I or iC I L L = r I H (5.11) @ 2 S Max @ a n N @ m M m M = @ H @ a n N or M T n N ;m M m M = r a n N H (5.12) @ 2 S gauge @ J @ K K = @ H @ J or N T J K K = r J H (5.13) @ 2 iS Sc h @ J @ K K = @ H @ J or iC J K K = r J H (5.14) whic h are of the Hamiltonian form = @ H =@ : The summation con v en tion is used throughout. These equations ma y b e written more compactly as M a = r H M T = r a H N = r H N T = r H iC = r H iC = r H (5.15)

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87 and can b e cast in to matrix form as 0BBBBBBBBBBBBBB@ iC 0 0 0 0 0 0 iC 0 0 0 0 0 0 0 0 M T 0 0 0 0 0 0 N T 0 0 M 0 0 0 0 0 0 N 0 0 1CCCCCCCCCCCCCCA 0BBBBBBBBBBBBBB@ _ a _ 1CCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBB@ @ H =@ @ H =@ @ H =@ a @ H =@ @ H =@ @ H =@ 1CCCCCCCCCCCCCCA ; (5.16) where the matrices M ; N ; and C and dened in ( 5.9 )-( 5.14 ). This symplectic form almost has the canonical structure of ( 4.25 ). In a basis of rank N ; the con tractions in v olving a and run to 3 N while the con tractions in v olving the remaining dynamical v ariables run to N : This is b ecause a and are spatial v ectors that ha v e ( x; y ; z )-comp onen ts whereas the remaining dynamical v ariables are scalars. With the c hoice of represen tation in ( 5.3 ) and the c hoice of basis in ( 5.4 ) all appro ximations are sp ecied. The equations of motion in ( 5.16 ) are the basis represen tation of the coupled Maxw ell-Scr odinger equations. They are automatically obtained b y applying the time-dep enden t v ariational principle to the Lagrangians ( 5.5 )-( 5.7 ). In the limit of a complete basis these equations are exact. The complex phase space that carries the asso ciated dynamics is endo w ed with the P oisson brac k et f F ; G g = 0BBBBBBBBBBBBBB@ @ F =@ @ F =@ @ F =@ a @ F =@ @ F =@ @ F =@ 1CCCCCCCCCCCCCCA T 0BBBBBBBBBBBBBB@ iC 0 0 0 0 0 0 iC 0 0 0 0 0 0 0 0 M T 0 0 0 0 0 0 N T 0 0 M 0 0 0 0 0 0 N 0 0 1CCCCCCCCCCCCCCA 1 0BBBBBBBBBBBBBB@ @ G=@ @ G=@ @ G=@ a @ G=@ @ G=@ @ G=@ 1CCCCCCCCCCCCCCA : (5.17)

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88 Ev en though the symplectic form is not canonical, its in v ersion is simple. The matrix elemen ts in in v olv e gaussian o v erlap in tegrals lik e h G I j G K i = R V h G I j x ih x j G K i d 3 x: 5.2 Maxw ell-Sc hr odinger Theory in a Real Basis As w as done previously eac h dynamical v ariable ma y b e expanded in to a complete basis of functions G K as Q ( x ; t ) = P K G K ( x ) q K ( t ) P ( x ; t ) = P K G K ( x ) p K ( t ) A k ( x ; t ) = P K G K ( x ) a k K ( t ) k ( x ; t ) = P K G K ( x ) k K ( t ) ( x ; t ) = P K G K ( x ) K ( t ) ( x ; t ) = P K G K ( x ) K ( t ) ; (5.18) where the index K runs o v er the basis and the index k runs o v er 1, 2, 3 or x; y ; z : Unlik e in ( 5.3 ), the co ecien ts in ( 5.18 ) that carry the dynamics are all real-v alued. In this basis, the real Lagrangian densities b ecome L Max = P KM 1 2 [( @ =@ a m M ) a m M ( @ =@ m M ) m M ] S Max H Max (5.19) L Sc h = P K 1 2 [( @ =@ q K ) q K ( @ =@ p K ) p K ] S Sc h H Sc h (5.20) L gauge = P KM 1 2 [( @ =@ K ) K ( @ =@ K ) K ] S gauge H gauge (5.21) with in tegrals S Max = R V A d 3 x S Sc h = R V P Qd 3 x S gauge = R V d 3 x: (5.22) Applying the calculus of v ariations to the ab o v e Lagrangians leads to the equations of motion: @ 2 S Max @ m M @ a n N a n N = @ H @ m M or M m M ;n N a n N = r m M H (5.23) @ 2 S gauge @ I @ L L = @ H @ I or N I L L = r I H (5.24)

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89 @ 2 S Sc h @ p I @ q L q L = @ H @ p I or C I L q L = r p I H (5.25) @ 2 S Max @ a n N @ m M m M = @ H @ a n N or M T n N ;m M m M = r a n N H (5.26) @ 2 S gauge @ J @ K K = @ H @ J or N T J K K = r J H (5.27) @ 2 S Sc h @ q J @ p K p K = @ H @ q J or C T J K p K = r q J H (5.28) whic h are of the Hamiltonian form = @ H =@ : These equations ma y b e written more compactly as M a = r H M T = r a H N = r H N T = r H C q = r p H C T p = r q H (5.29) and can b e cast in to matrix form as 0BBBBBBBBBBBBBB@ 0 0 0 M T 0 0 0 0 0 0 N T 0 0 0 0 0 0 C T M 0 0 0 0 0 0 N 0 0 0 0 0 0 C 0 0 0 1CCCCCCCCCCCCCCA 0BBBBBBBBBBBBBB@ a _ q _ p 1CCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBB@ @ H =@ a @ H =@ @ H =@ q @ H =@ @ H =@ @ H =@ p 1CCCCCCCCCCCCCCA ; (5.30) where the matrices M ; N ; and C and dened in ( 5.23 )-( 5.28 ). Again they are the basis represen tation of the coupled Maxw ell-Sc hr odinger equations of motion.

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90 The real phase space that carries the asso ciated dynamics is endo w ed with the P oisson brac k et f F ; G g = 0BBBBBBBBBBBBBB@ @ F =@ a @ F =@ @ F =@ q @ F =@ @ F =@ @ F =@ p 1CCCCCCCCCCCCCCA T 0BBBBBBBBBBBBBB@ 0 0 0 M T 0 0 0 0 0 0 N T 0 0 0 0 0 0 C T M 0 0 0 0 0 0 N 0 0 0 0 0 0 C 0 0 0 1CCCCCCCCCCCCCCA 1 0BBBBBBBBBBBBBB@ @ G=@ a @ G=@ @ G=@ q @ G=@ @ G=@ @ G=@ p 1CCCCCCCCCCCCCCA : (5.31) Ev en though the symplectic form is not canonical, its in v ersion is simple once again. The matrix elemen ts in in v olv e gaussian o v erlap in tegrals lik e h G I j G K i = R V h G I j x ih x j G K i d 3 x: 5.2.1 Ov erview of Computer Program The equations of motion ( 5.30 ) are co ded in F or tran 90. The computer program is called Electron Nuclear Radiation Dynamics or ENRD. Eac h matrix elemen t in the symplectic form and in the forces is p erformed analytically The program is rexible enough to handle a rank N basis of s -gaussians, eac h with an adjustable width and an arbitrary cen ter. A n umerical solution to ( 5.30 ) is determined once the initial v alue data is sp ecied for b : The forces @ H =@ a are constructed from this data. The symplectic form ab is then in v erted with the LAP A CK [ 91 ] subroutine DGESVX, whic h is the exp ert driv er for the AX = B solv er DGESV. This establishes a rst order system of dieren tial equations of the form b = 1 ab @ H =@ a whic h ma y b e solv ed, for example, with an Euler stepping metho d. That is b ( t + t ) = b ( t ) + ( t ) b ( t ) = b ( t ) + ( t )[ 1 ab @ H =@ a ]( t ) : (5.32) In practice, the Euler metho d is not accurate enough so the more sophisticated RK4 metho d [ 92 ] is implemen ted in the co de. The equations of motion are adv anced

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91 at a xed stepsize of 10 3 au. F or t ypical basis function widths and cen ters, the estimated condition n um b er rep orted b y DGESVX is ab out 30. Lastly it should b e p oin ted out that the equations of motion ( 5.30 ) are n umerically implemen ted in terms of the electric eld E 4 c rather than the momen tum : It w as found that w orking in terms of this new (scaled) co ordinate pro vides a more balanced set of dynamical equations. Nev ertheless, the electromagnetic radiation is still quite small compared to the dynamics of the matter. An o v erview of the ENRD program is presen ted in Figure 5{1

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92 PSfrag replacemen ts enrd.f90 RHS EM.f90 RHS SH.f90 Create library for four index Create library for v ector Input de ck. RHS RHS Maxw ell. RHS Gauge. matrix elemen ts. DERIVS.f90 MAXENER GY.f90 GAENER GY.f90 SCHENER GY.f90 Maxw ell energy Gauge energy .@ H @ ;@ H @ A @ H @ ;@ H @ @ H @ P;@ H @ Q Sc hr odinger energy Sc hr odinger. Build forces@ H @ on RHS. RHS GA.f90 RIRJ.f90 RK4.f90 Runge-Kutta 4 ODE solv er. ( t + t ) = ( t ) + ( t ) ( t ) In v ert = @ H =@ to get = 1@ H =@ : METRIC.f90 Build symplectic form : Call LAP A CK routine DGESVX. argumen ts. Symplectic form : Build v ector argumen ts. Call in tegrator. W rite ( t + t ) to PS.dat Compute h Mj ( t ) i : Compute h Mj ( t ) i : Compute h Mj ( t ) i : Figure 5{1: Sc hematic o v erview of ENRD computer program.

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93 5.2.2 Stationary States: s and p -W a v es The ENRD program w as rst tested with a stationary state of the h ydrogen atom. In a basis of six gaussians an s -w a v e w as constructed as w ell as the corresp onding basis represen tation of the Coulom bic scalar p oten tial and the Coulom bic electric eld. In fact, an y spherically symmetric distribution of c harge along with the corresp onding Coulom b elds w ould suce. This delicate balance of c harges and elds pro v ed to b e a stationary state of the com bined system. No electromagnetic radiation w as pro duced. The total c harge q = R V ( x ) d 3 x remained constan t. A p x -w a v e and its asso ciated Coulom bic elds w ere also created in the same basis. This again is a stationary state. 5.2.3 Nonstationary State: Mixture of s and p -W a v es After iden tifying some stationary states, a nonstationary state that is a mixture of s and p x -w a v es w as constructed in the same rank six basis. Both the Coulombic scalar p oten tial and the Coulom bic electric eld that are asso ciated with this c harge distribution w ere created as w ell. Electromagnetic radiation w as pro duced as the electron oscillated b et w een stationary states. Energy momen tum, and angular momen tum w ere exc hanged b et w een the electron and the electromagnetic eld. It w as sho wn that the total energy and total Hamiltonian are conserv ed to four decimal places. The total c harge q = R V ( x ) d 3 x remained constan t. The phase space con tours for the electromagnetic eld, matter eld, and gauge eld are presen ted in Figures 5{2 5{3 and 5{4 resp ectiv ely 5.2.4 F ree Electro dynamics Lastly a free electromagnetic eld w as constructed. In this case no c harge w as created. Energy momen tum, and angular momen tum w ere exc hanged only b et w een the electromagnetic and gauge degrees of freedom. The total energy remained constan t.

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94 PSfrag replacemen ts a k K ( t ) k K ( t ) 0.5 -0.5 0 0 10 -10 20 -20 Figure 5{2: Phase space con tour for the co ecien ts of the v ector p oten tial A and its momen tum : PSfrag replacemen ts q K ( t ) p K ( t ) 0 0 5 5 -5 -5Figure 5{3: Phase space con tour for the co ecien ts of the real-v alued Sc hr odinger eld Q and its momen tum P :

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95 PSfrag replacemen ts K ( t ) K ( t ) 0.0005 -0.0005 0 0 10 -10 20 -20 -30 Figure 5{4: Phase space con tour for the co ecien ts of the scalar p oten tial and its momen tum : 5.2.5 Analysis of Solutions in Numerical Basis The solutions ( t ) of the equations of motion ( 5.30 ) are further analyzed b y expansion in to the basis eigenstates M : The Sc hr odinger eigenstates are found b y diagonalizing the time-indep enden t Sc hr odinger equation HC = SC ; (5.33) where H is the basis represen tation of the Hamiltonian H = r 2 = 2 m + V ; C is the matrix of basis expansion co ecien ts, S is the basis o v erlap matrix, and is the matrix of energy eigen v alues. Similarly the Maxw ell eigenstates are found b y diagonalizing the free w a v e equation r 2 =c 2 = 0 ; where can b e the scalar p oten tial or an y comp onen t of the v ector p oten tial A : F ourier in v ersion of the free w a v e equation results in c 2 r 2 ~ = 2 ~ ; where is the frequency In a basis this equation turns in to the matrix equation HC = 2 SC ; (5.34)

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96 where H is the basis represen tation of the Hamiltonian-lik e quan tit y H = c 2 r 2 ; C is the matrix of basis expansion co ecien ts, S is the basis o v erlap matrix, and 2 is the matrix of frequencies squared. Recall that energy is related to frequency b y E = ~ ; so that in atomic units energy is equiv alen t to frequency Both of these basis equations ( 5.33 ) and ( 5.34 ) are recognized as b elonging to the generalized eigen v alue problem A = B ; whic h can b e in v erted with the LAP A CK routine DSYGV. The ENRD program emplo ys DSYGV to solv e b oth ( 5.33 ) and ( 5.34 ) for their corresp onding eigen v alues M and eigen v ectors M : With the eigen v ectors M ; the ev olving state v ector ( t ) can b e expanded according to j ( t ) i = P M j M ih M j ( t ) i ; (5.35) where C M ( t ) h M j ( t ) i are the basis expansion co ecien ts. The real and imaginary parts of the Sc hr odinger co ecien ts for a sup erp osition of s and p x -w a v es are plotted v ersus time in Figures 5{5 and 5{6 resp ectiv ely The squares of these co ecien ts are plotted v ersus time in Figure 5{7 Notice in Figures 5{5 and 5{6 that there are three frequencies in v olv ed in the dynamics, whic h corresp ond to excitations of the s -, p x -, and d y 2 z 2 -w a v es. Figure 5{7 suggests that the electron deca ys from p x to s in under 10 au of time. Ho w ev er, due to the nite size of the basis, the electron is excited bac k to the p x -state as the electromagnetic elds rerect o of the articial basis b oundaries. Lastly the phase space con tour of the Sc hr odinger co ecien ts are presen ted in Figure 5{8

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97 PSfrag replacemen ts t 0 0 10 20 30 Re h M j ( t ) i s p x p y p z d x 2 d y 2 z 2 1 1Figure 5{5: Real part of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es. PSfrag replacemen ts t 0 0 10 20 30 Im h M j ( t ) i s p x p y p z d x 2 d y 2 z 2 1 1Figure 5{6: Imaginary part of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es.

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98 PSfrag replacemen ts t P M jh M j ( t ) ij 2 s p x jh M j ( t ) ij 2 s px py pz dx2 dy2 z2 0 0 10 20 30 0 : 5 0 : 2 0 : 4 0 : 6 0 : 8 10 20 30 0 : 001 0 : 001 1 1 Re h Nj ( t ) i Im h M j ( t ) i qK( t ) p K ( t ) Figure 5{7: Probabilit y for the electron to b e in a particular basis eigenstate. PSfrag replacemen ts t s p x p y p z d x 2 d y 2 z 2 0 0 10 20 25 30 0 : 5 0 : 5 0 : 5 0 : 5 10 20 30 0 : 001 0 : 001 1 1 1 1 Re h M j ( t ) i Im h M j ( t ) i qK( t ) p K ( t ) Figure 5{8: Phase space of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es.

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99 5.3 Symplectic T ransformation to the Coulom b Reference Recall the basis represen tation of the Maxw ell-Sc hr odinger equations of motion in ( 5.30 ). They are 0BBBBBBBBBBBBBB@ 0 0 0 M T 0 0 0 0 0 0 N T 0 0 0 0 0 0 C T M 0 0 0 0 0 0 N 0 0 0 0 0 0 C 0 0 0 1CCCCCCCCCCCCCCA 0BBBBBBBBBBBBBB@ a _ q _ p 1CCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBB@ @ H =@ a @ H =@ @ H =@ q @ H =@ @ H =@ @ H =@ p 1CCCCCCCCCCCCCCA ; ( 5.30 ) where M KM @ 2 h j A i @ K @ a M N KM @ 2 h j i @ K @ M C KM @ 2 h P j Q i @ p K @ q M (5.36) and where the in tegrals hji in v olv e only gaussian functions. In analogy to the transformation ( T 1 ) T in ( 4.44 ) that w as dened on the v ector elds @ =@ in function space, a basis represen tation of ( T 1 ) T can b e made. This basis represen tation is dened in terms of the co ecien ts according to 0BBBBBBBBBBBBBB@ @ =@ ~ a @ =@ ~ @ =@ ~ q @ =@ ~ @ =@ ~ @ =@ ~ p 1CCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBB@ @ a=@ ~ a 0 0 0 0 0 0 @ =@ ~ 0 0 0 0 0 @ =@ ~ q @ q =@ ~ q @ =@ ~ q 0 0 0 0 0 @ =@ ~ 0 0 0 0 0 0 @ =@ ~ 0 0 @ =@ ~ p 0 @ =@ ~ p 0 @ p=@ ~ p 1CCCCCCCCCCCCCCA 0BBBBBBBBBBBBBB@ @ =@ a @ =@ @ =@ q @ =@ @ =@ @ =@ p 1CCCCCCCCCCCCCCA (5.37)

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100 so that the symplectic form in ( 5.30 ) transforms as 0BBBBBBBBBBBBBB@ 0 0 0 M T 0 0 0 0 0 0 N T 0 0 0 0 0 0 C T M 0 0 0 0 0 0 N 0 0 0 0 0 0 C 0 0 0 1CCCCCCCCCCCCCCA 0BBBBBBBBBBBBBB@ 0 0 M ~ A ~ Q M ~ A ~ 0 M ~ A ~ P 0 0 0 0 N ~ ~ 0 M ~ Q ~ A 0 0 0 N ~ Q ~ C ~ Q ~ P M ~ ~ A 0 0 0 0 0 0 N ~ ~ N ~ ~ Q 0 0 N ~ ~ P M ~ P ~ A 0 C ~ P ~ Q 0 N ~ P ~ 0 1CCCCCCCCCCCCCCA ; (5.38) where the new matrix elemen ts are: M ~ X ~ Y @ @ ~ X M @ a @ ~ Y = @ @ ~ X @ 2 h ~ + C j ~ A i @ @ a @ a @ ~ Y = @ 2 h ~ + C j ~ A i @ ~ X @ ~ Y N ~ X ~ Y @ @ ~ X N @ @ ~ Y = @ @ ~ X @ 2 h j ~ + C i @ @ @ @ ~ Y = @ 2 h ~ j ~ + C i @ ~ X @ ~ Y C ~ X ~ Y @ p @ ~ X C @ q @ ~ Y = @ p @ ~ X @ 2 h P j Q i @ p@ q @ q @ ~ Y = @ 2 h ~ P j Q i @ ~ X @ ~ Y (5.39) for X and Y an arbitrary dynamical v ariable. The remaining elemen ts are determined b y transp osition. Again the extra terms in ~ add new time-dep enden t couplings to the theory These new terms can all b e p erformed analytically The

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101 resulting equations of motion are 0BBBBBBBBBBBBBB@ 0 0 M ~ A ~ Q M ~ A ~ 0 M ~ A ~ P 0 0 0 0 N ~ ~ 0 M ~ Q ~ A 0 0 0 N ~ Q ~ C ~ Q ~ P M ~ ~ A 0 0 0 0 0 0 N ~ ~ N ~ ~ Q 0 0 N ~ ~ P M ~ P ~ A 0 C ~ P ~ Q 0 N ~ P ~ 0 1CCCCCCCCCCCCCCA 0BBBBBBBBBBBBBB@ ~ a ~ ~ q ~ ~ ~ p 1CCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBB@ @ H =@ ~ a @ H =@ ~ @ H =@ ~ q @ H =@ ~ @ H =@ ~ @ H =@ ~ p 1CCCCCCCCCCCCCCA : (5.40) 5.3.1 Numerical Implemen tation Recall that the forces app earing in the canonical transformed equations ( 4.57 ) could b e simplied b y substituting these equations among themselv es. As a result ~ ! 0BBBBBBBBBBBBBB@ 0 0 M ~ A ~ Q M ~ A ~ 0 M ~ A ~ P 0 0 0 0 N ~ ~ 0 0 0 0 0 0 C ~ Q ~ P M ~ ~ A 0 0 0 0 0 0 N ~ ~ N ~ ~ Q 0 0 N ~ ~ P 0 0 C ~ P ~ Q 0 0 0 1CCCCCCCCCCCCCCA : (5.41) Note that ( 5.41 ) is not a symplectic form. After making this substitution, the Hamiltonian structure is lost. Ho w ev er, the n umerical implemen tation is greatly facilitated with the simplied equations ( 4.57 ) instead of those in ( 4.56 ). Since the ENRD program do es not rely on a symplectic in tegrator sc heme to adv ance the dynamics, the symplectic structure is not n umerically imp ortan t an yw a y The equations of motion ( 4.57 ) has b een added to the ENRD co de. The Coulom b reference can b e con v enien tly turned on or o (resulting in ( 4.32 )) with an optional rag. As b efore, the program is rexible enough to handle a rank N basis of s -gaussians, eac h with an adjustable width and an arbitrary cen ter. A solution

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102 to ( 4.57 ) ma y b e obtained once the initial v alue data is sp ecied for ~ b : The forces @ H =@ ~ a are constructed from this data. The new terms app earing on left hand side of ( 4.57 ) are co ded analytically Notice that these terms mak e up the elemen ts of a matrix that is not a symplectic form. Nev ertheless, the resulting matrix equations are in tegrated with the same RK4 stepping metho d and the DGESVX subroutine of LAP A CK. F or t ypical basis function widths and cen ters, the condition n um b er rep orted b y DGESVX is on the order of one thousand. 5.3.2 Stationary States: s and p -W a v es The ENRD program with the Coulom b reference w as rst tested with a stationary state of the h ydrogen atom. In a basis of six gaussians an s -w a v e w as constructed. The corresp onding basis represen tation of the Coulom bic scalar p otential and the Coulom bic electric eld w ere not needed. All Coulom bic prop erties are treated analytically once the Coulom b reference is c hosen. Again, it w as found that an y spherically symmetric distribution of c harge will suce to pro duce an s -w a v e that is a stationary state of the com bined system. No electromagnetic radiation w as pro duced. The total c harge q = R V ( x ) d 3 x remained constan t. A p x -w a v e w as also created in the same basis. This again w as a stationary state of the com bined system. 5.3.3 Nonstationary State: Mixture of s and p -W a v es After iden tifying some stationary states, a nonstationary state that is a mixture of s and p x -w a v es w as constructed in the same rank six basis. Both the Coulom bic scalar p oten tial and the Coulom bic electric eld that are asso ciated with this c harge distribution w ere done analytically b y the canonical transformation to the Coulom b reference. Electromagnetic radiation w as pro duced as the electron oscillated b et w een stationary states. Energy momen tum, and angular momen tum w ere exc hanged b et w een the electron and the electromagnetic eld. It w as sho wn that the total energy and total Hamiltonian are conserv ed to t w o decimal places. The total c harge q = R V ( x ) d 3 x remained constan t.

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103 5.3.4 F ree Electro dynamics Lastly a free electromagnetic eld w as constructed. In this case no c harge w as created. Energy momen tum, and angular momen tum w ere exc hanged only b et w een the electromagnetic and gauge degrees of freedom. The total energy remained constan t. 5.3.5 Analysis of Solutions in Coulom b Basis As done previously the ev olving state v ector ~ ( t ) in the Coulom b basis is expanded in terms of the stationary eigen basis ~ M M according to j ~ ( t ) i = P M j ~ M ih ~ M j ~ ( t ) i : ( 5.35 ) As b efore, the real and imaginary parts of the Sc hr odinger co ecien ts for a sup erp osition of s and p x -w a v es are plotted v ersus time in Figures 5{9 and 5{10 resp ectiv ely The squares of these co ecien ts are plotted v ersus time in Figure 5{11 Notice in Figures 5{9 and 5{10 that there are again three frequencies in v olv ed in the dynamics, whic h corresp ond to excitations of the s -, p x -, and d y 2 z 2 -w a v es. Figure 5{11 suggests that the electron deca ys from p x to s in under 15 au of time. Ho w ev er, due to the same aforemen tioned basis eects, the electron oscillates b et w een the s and p x -states. Lastly the phase space con tour of the Sc hr odinger co ecien ts are presen ted in Figure 5{12 5.4 Asymptotic Radiation It has b een demonstrated that the dynamics of the h ydrogen atom's electron in the presence of the electromagnetic eld w as quasip erio dic. This unph ysical b eha vior is due to the fact the electromagnetic radiation pro duced b y the electron cannot escap e to innit y and carry a w a y energy momen tum, and angular momentum. Rather, the radiation rerects o of the articial b oundaries of the nite spatial basis and reexcites the electron.

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104 PSfrag replacemen ts t 0 0 10 20 30 Re h M j ( t ) i s p x p y p z d x 2 d y 2 z 2 1 1Figure 5{9: Real part of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es. PSfrag replacemen ts t 0 0 10 20 30 Im h M j ( t ) i s p x p y p z d x 2 d y 2 z 2 1 1Figure 5{10: Imaginary part of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es.

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105 PSfrag replacemen ts t P M jh M j ( t ) ij 2 s p x jh M j ( t ) ij 2 s px py pz dx2 dy2 z2 0 0 10 20 30 0 : 5 0 : 2 0 : 4 0 : 6 0 : 8 10 20 30 0 : 001 0 : 001 1 1 Re h Nj ( t ) i Im h M j ( t ) i qK( t ) p K ( t ) Figure 5{11: Probabilit y for the electron to b e in a particular basis eigenstate. PSfrag replacemen ts t s p x p y p z d x 2 d y 2 z 2 0 0 10 20 25 30 0 : 5 0 : 5 0 : 5 0 : 5 10 20 30 0 : 001 0 : 001 1 1 1 1 Re h M j ( t ) i Im h M j ( t ) i qK( t ) p K ( t ) Figure 5{12: Phase space of the Sc hr odinger co ecien ts C M ( t ) h M j ( t ) i ; where ( t ) is a sup erp osition of s and p x -w a v es.

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106 The asymptotic problem, b e it electromagnetic radiation or free (ionized) electrons, has p osed a dicult n umerical c hallenge. F ree electromagnetic radiation in v acuum do es not spread in time, since there is no disp ersion, but do es tra v el at the sp eed of ligh t c 137 au. Ho w ev er, the v elo cit y of the sources of c harge and curren t, e.g. the electron in the h ydrogen atom, is on the order of 1 au. This drastically dieren t v elo cit y scale mak es a n umerical description of the time-dep enden t theory in direct space quite demanding. On the other hand, the description of the free nonrelativistic electron is made dicult b y a com bination of its large v elo cit y v ( v < c ), the spreading of its w a v epac k et, and the rapid oscillation of its phase. Ev en in v acuum, the Sc hr odinger equation is disp ersiv e so that the electronic w a v epac k et width gro ws prop ortionally with time and its phase gro ws quadratically with the distance from the cen ter of the w a v epac k et. Sev eral tec hniques ha v e b een dev elop ed to partially treat these problems. In 1947, Wigner and Eisen bud [ 93 ] dev elop ed the R -matrix metho d, whic h pro vides a tec hnique for matc hing the solutions on some surface separating the inner b ound state region and outer scattering state region. More recen tly masking functions, rep etitiv e pro jection and complex rotation metho ds, and Siegert pseudostates are common theoretical to ols. These tec hniques are discussed b y Y oshida, W atanab e, Reinhold, and Burgd orfer in [ 94 ] and b y T olstikhin, Ostro vsky and Nak am ura in [ 95 ]. A scaling transformation metho d that eliminates the rapid phase v ariation and w a v epac k et expansion and requires no matc hing at innit y has b een presen ted b y Sidky and Esry in [ 96 ]. Lastly McCurdy and collab orators [ 97 { 99 ] ha v e eectiv ely implemen ted an exterior complex scaling metho d [ 100 ] in the time-indep enden t form ulation of scattering theory The exterior complex scaling metho d maps all co ordinates b ey ond a certain radius to a con tour that is rotated b y some xed angle in to the complex plane. This tec hnique damps all purely outgoing scattered w a v es to zero exp onen tially whic h p ermits a n umerical treatmen t on a nite domain or grid.

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107 PSfrag replacemen ts Lo cal basis Asymptotic basis t ( t ) Figure 5{13: Sc hematic picture of the lo cal and asymptotic basis prop osed for the description of electromagnetic radiation and electron ionization. The amplitude from the asymptotic basis is dump ed in to the free eld ; whic h acts as a storage tank for energy and probabilit y A form ulation of the asymptotic n umerical problem that falls more in line with the canonical treatmen t presen ted in this dissertation w ould b egin at the Lagrangian lev el with a Lagrangian of the form L = L ENRD + L coupling + L free : (5.42) The ENRD Lagrangian L ENRD w ould b e the Maxw ell-Sc hr odinger Lagrangian from ( 1.17 ) and ( 1.18 ). The dynamics of this system w ould b e describ ed b y t w o dieren t t yp es of basis functions. As pictured in Figure 5{13 the atomic or molecular system w ould ha v e a lo cal basis represen tation in terms of real gaussian basis functions of the form G K ( x ) = G K ( x ) = N K exp ( ` K [ x r K ] 2 ) : ( 5.4 ) F urther a w a y a set of complex basis functions of the form

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108 G k ( x ) = P l m C l m Y l m ( ^ x ) e ik r r exp ( ar 2 ) (5.43) w ould b e used, where the w a v ev ector magnitude k = =c could b e c hosen to lie in some range k min k k max and most lik ely only a few l w ould b e necessary These complex basis functions will require the calculation of new matrix elemen ts. The free Lagrangian L free w ould b e the free particle Lagrangian i f i r i r g = 2 m or the free eld Lagrangian @ @ : The solutions of the free equations of motion deriv ed from these free Lagrangians are kno wn analytically and are of the form exp ( i [ k x t ]) : The coupling Lagrangian should b e a Loren tz scalar that is made up of a certain com bination or pro duct of dynamical v ariables of L ENRD and of L free : If amplitude is put in to the co ecien ts of the asymptotic basis functions G k ; then the amplitude will transfer to the free solutions or : This amplitude will pro vide an initial condition for the free elds, thereb y dening or throughout spacetime. The free elds will store the energy and probabilit y (and momen tum and angular momen tum) radiated at innit y whic h is needed to main tain the v arious conserv ation la ws. 5.5 Proton Dynamics in a Real Basis The previous complex Sc hr odinger Lagrangians ma y b e written in real form b y taking the electronic w a v efunction = [ Q + iP ] = p 2 and the protonic w a v efunction n = [ U + iW ] = p 2 : In terms of these real dynamical v ariables the Hamiltonian densit y b ecomes H = [ 4 c ] 2 + [ r A ] 2 [4 c ] 2 8 c r c r A + [ r Q + q A P =c ] 2 + [ r P + q A Q=c ] 2 4 m q + q Q 2 + P 2 2 + [ r U + q A W =c ] 2 + [ r W + q A U =c ] 2 4 m q + q U 2 + W 2 2 (5.44)

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109 As in ( 5.3 ), eac h of the dynamical v ariables ma y b e expanded in to a basis. In particular, the real and imaginary comp onen ts of and n are expanded as Q ( x ; t ) = P K G K ( x ) q K ( t ) U ( x ; t ) = P K G K ( x ) u K ( t ) P ( x ; t ) = P K G K ( x ) p K ( t ) W ( x ; t ) = P K G K ( x ) w K ( t ) : (5.45) The basis functions G K are c hosen to b e simple s -gaussians. The Hamilton equations of motion asso ciated with these real dynamical v ariables are 0BBBBBBBBBBBBBBBBBBBB@ 0 0 0 M T 0 0 0 0 0 0 0 N T 0 0 0 0 0 0 0 C T 0 0 0 0 0 0 0 K T M 0 0 0 0 0 0 0 N 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 K 0 0 0 1CCCCCCCCCCCCCCCCCCCCA 0BBBBBBBBBBBBBBBBBBBB@ a _ q u _ p w 1CCCCCCCCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBBBBBBBB@ @ H =@ a @ H =@ @ H =@ q @ H =@ u @ H =@ @ H =@ @ H =@ p @ H =@ w 1CCCCCCCCCCCCCCCCCCCCA ; ( 5.30 ) where M KM @ 2 h j A i @ K @ a M N KM @ 2 h j i @ K @ M C KM @ 2 h P j Q i @ p K @ q M K KM @ 2 h W j U i @ w K @ u M (5.46) and where the in tegrals hji in v olv e only gaussian functions.

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CHAPTER 6 CONCLUSION Nonp erturbativ e analytical and n umerical metho ds for the solution of the nonlinear Maxw ell-Sc hr odinger equations ha v e b een presen ted including the complete coupling of b oth systems. The theory b egins b y applying the calculus of v ariations to the Maxw ell and Sc hr odinger Lagrangians together with a gauge xing term for the Lorenz gauge. Within the Hamiltonian or canonical prescription, this yields a set of rst order dieren tial equations in time of the form ab b = @ H =@ a : (6.1) These Maxw ell-Sc hr odinger equations are closed when the Sc hr odinger w a v efunction is c hosen as a source for the electromagnetic eld and the electromagnetic eld acts bac k up on the w a v efunction. Moreo v er, this system of equations forms a w elldened initial v alue problem. That is, the en tire dynamics is kno wn in principle once the initial v alues for eac h of the dynamical v ariables are sp ecied. The resulting dynamics enjo ys conserv ation of energy momen tum, angular momen tum, and c harge b et w een the matter and the electromagnetic eld. In practice, the Maxw ell-Sc hr odinger equations are represen ted in a nite basis of gaussian functions G K ( x ) and solv ed n umerically That is, eac h dynamical v ariable is expanded in this basis according to ( x ; t ) = P K G K ( x ) K ( t ) ; (6.2) where the time-dep enden t co ecien ts K ( t ) carry the dynamics. As a result, a hierarc h y of appro ximate equations of motion are generated that basis-represen t the exact Maxw ell-Sc hr odinger equations and can b e made systematically more and more 110

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111 accurate b y enric hing the basis. In the limit of a complete basis, these appro ximate equations w ould b e exact since the gaussian functions span L 2 : The basis represen tation of the Maxw ell-Sc h odinger equations of motion has b een n umerically implemen ted in a F or tran 90 computer program. This program allo ws for an arbitrary rank basis of s -gaussians of v arying widths and cen ters. It has b een used to explore the dynamics of the h ydrogen atom in teracting with the electromagnetic eld. In particular, stationary states of the com bined atomeld system ha v e b een constructed as w ell as nonstationary states that radiate. This radiation carries a w a y energy momen tum, and angular momen tum from the h ydrogen atom. A series of plots are presen ted to do cumen t the radiativ e deca y of h ydrogen's electron from a sup erp osition of s and p x states to the s ground state. In order to impro v e n umerical con v ergence, a canonical transformation w as p erformed on the Maxw ell-Sc hr odinger equations to isolate the Coulom bic or electrostatic con tribution to the scalar p oten tial C and electric eld E C : This p ortion of the elds can b e p erformed analytically once the source is sp ecied b y solving the P oisson equation r 2 C = 4 and then calculating E C = r C : By remo ving the burden of describing b oth the Coulom bic and radiativ e con tributions to the electro dynamics, the eorts of the basis are fo cused en tirely on the description of the radiation. The canonical transformed equations of motion ha v e b een represen ted in a gaussian basis as done previously and ha v e b een added to the existing F or tran 90 computer program. With an optional rag the Coulom b reference can b e used. Otherwise the ra w n umerical basis is used b y default. As b efore, a series of plots are presen ted to do cumen t the dynamics of the h ydrogen atom in teracting with the electromagnetic eld. The results in b oth cases are analyzed. The w ork presen ted in this dissertation is particularly applicable to ph ysical situations where the dynamics of the sources of c harge and curren t o ccurs on the same timescale as the dynamics of the electromagnetic eld. In these situations, adiabatic

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112 and p erturbativ e approac hes ma y b e insucien t to describ e the strongly coupled matter-eld dynamics. P ossible applications of the Maxw ell-Sc hr odinger theory lie in photon-electron-phonon dynamics in semiconductor quan tum w ells [ 10 ], sp on taneous emission in cold atom collisions [ 11 12 ], atom-photon in teraction in single atom laser ca vities [ 14 15 ], and photon-exciton dynamics in ruorescen t p olymers [ 16 ].

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APPENDIX A GA UGE TRANSF ORMA TIONS A.1 Gauge Symmetry of Electro dynamics The basic equations of electro dynamics [ 9 ] are: r E = 4 r B = 4 c J + E c r B = 0 r E + B c = 0 ; (A.1) where and J are the c harge and curren t densities. These Maxw ell equations ma y b e rewritten in terms of the scalar and v ector p oten tials and A as r 2 + r A c = 4 (A.2a) r 2 A A c 2 r h r A + c i = 4 c J (A.2b) b y dening the electric eld E = A =c r and the magnetic eld B = r A : Maxw ell's equations written in either eld or p oten tial forms are in v arian t under the gauge transformation 0 = G c A A 0 = A + r G; (A.3) where G is an arbitrary and w ell-b eha v ed function called the gauge function or gauge generator. It is said that Maxw ell's equations enjo y the full gauge freedom. Ho w ev er, a particular gauge ma y b e c hosen with an appropriate c hoice of the gauge function. F or example, a gauge transformation can alw a ys b e made on the p oten tials so that the Lorenz gauge condition is satised. That is 0 = r A 0 + 0 c = r A + r 2 G + c G c (A.4) 113

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114 implies that the Lorenz gauge function satises the equation G = [ r A + =c ] : This c hoice of gauge function leads to the manifestly Loren tz in v arian t equations of motion: r 2 c 2 = 4 (A.5a) r 2 A A c 2 = 4 c J : (A.5b) Another c hoice is the Coulomb ( r adiation ) gauge whic h leads to the equations of motion: r 2 = 4 (A.6a) r 2 A T A T c 2 = 4 c J T (A.6b) with the gauge function satisfying r 2 G = r A : The Coulom b gauge v ector p oten tial A T is the transv erse pro jection of the Lorenz gauge v ector p oten tial A ; as is the curren t J T : That is A aT ( x ; t ) = h ab @ a @ b r 2 i A b ( x ; t ) = R V d 3 k (2 ) 3 e i k x h ab k a k b k 2 i ~ A b ( k ; t ) ; (A.7) where P ab T = ab @ a @ b = r 2 is the tr ansverse pr oje ction op er ator (see App endix C ) and 1 = r 2 is shorthand for the Green's function R V d 3 x 0 [ 4 j x x 0 j ] 1 of the Laplacian op erator (see App endix B ), where ( r 2 = r 2 ) g = g for g a w ell-b eha v ed function. More precisely 1 r 2 r 2 g ( x ) = R V d 3 x 0 g ( x 0 ) r 0 2 1 4 j x x 0 j = g ( x ) : (A.8) The dynamical radiation elds asso ciated with A T are almost separated from the instan taneous or static elds asso ciated with ; as seen in ( A.6 ). A closer examination of the transv erse pro jection op erator P ab T will sho w that ev en A T con tains instantaneous comp onen ts. That these instan taneous eects exactly cancel b et w een A T

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115 and to pro duce the causal E and B is one of the often misundersto o d prop erties of the Coulom b gauge [ 101 ]. Notice that the Lorenz gauge equations of motion ( A.5 ) enjo y a limited gauge freedom kno wn as the residual gauge symmetry In other w ords, within the Lorenz gauge there is still a limited family of gauge transformations that can b e made on the p oten tials that lea v e them in the Lorenz gauge. Suc h gauge generators satisfy the homogeneous equation G = 0 : Similarly there is residual gauge freedom left in the Coulom b gauge. That is, there is a limited class of gauge transformations that can b e made on ( A.6 ) that lea v e them unc hanged as w ell. Suc h gauge generators satisfy the homogeneous equation r 2 G = 0 : Man y other c hoices of gauge function are p ossible, eac h leading to a dieren t gauge. A particular gauge is often c hosen in accordance with a giv en ph ysical situation so as to simplify the asso ciated mathematics. See Cohen-T annoudji et. al. [ 30 ] for a discussion of other gauges p ertinen t in the con text of atomic and molecular ph ysics. A.2 Gauge Symmetry of Quan tum Mec hanics In addition to the electro dynamic gauge symmetries of ( A.3 ), quan tum mec hanics exhibits the additional symmetry 0 = e iq G=c : (A.9) The coupled system ( 3.20 ) and ( A.2 ) are in v arian t under gauge transformations ( A.3 ) and ( A.9 ) with the same gauge function G: While there are other gauge inv arian t coupling sc hemes, the minimal coupling prescription em b o died in ( 3.20 ) is the simplest.

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APPENDIX B GREEN'S FUNCTIONS The Gr e en 's function or fundamen tal solution G asso ciated with the inhomogeneous (partial) dieren tial equation Lu ( k ) = h ( k ) satises the equation LG ( k ; k 0 ) = ( n ) ( k k 0 ) (B.1) where L is a dieren tial op erator, u and h are C 1 and L 2 \ L 1 functions, and is the Dirac -function dened b elo w. The inhomogeneous term h is often called a source for u: In this language, the Green's function G ( k ; k 0 ) = G ( k 0 ; k ) is a solution of the dieren tial equation corresp onding to a p oin t-lik e source, i.e. the -function. Lo osely sp eaking the Green's function is the in v erse of the op erator L: The particular solution u of the dieren tial equation Lu = h ma y b e obtained in principle b y in tegration against G o v er all of space. That is u ( k ) = R d ( n ) k 0 h ( k 0 ) G ( k 0 ; k ) (B.2) whic h along with the solution of Lu = 0 constitutes the full solution. In other w ords, the dieren tial equation Lu = h has b een transformed in to an in tegral equation in whic h the Green's function is the k ernel. Substitution of this solution in to the dieren tial equation leads to h ( k ) = L R d ( n ) k 0 h ( k 0 ) G ( k 0 ; k ) = R d ( n ) k 0 h ( k 0 ) ( n ) ( k 0 k ) : (B.3) A brief discussion of the -function is presen ted in the next section. With this kno wledge, the Green's functions for the Laplacian r 2 and the w a v e op erator @ 2 = 2 = @ 2 =@ ( ct ) 2 r 2 are deriv ed. 116

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117 B.1 The Dirac -F unction The Dirac -function ma y b e dened in n -dimensions b y the v olume in tegral R d ( n ) k (2 ) n 1 e i k x = ( n ) ( x ) ; (B.4) where k x is the Euclidean scalar pro duct. This in tegral is just the F ourier in v erse of the unit 1. Notice that the -function is not a function but rather it is a distribution whic h is zero ev erywhere except at x = 0 : It ma y b e used as an in tegration k ernel to \pluc k out" the v alue of a function at a particular p oin t. F or example g ( x ) = R d ( n ) y g ( y ) ( n ) ( y x ) ; (B.5) where ( n ) ( y x ) = ( n ) ( x y ) : A particular c hoice for the function g suc h as g ( y ) = 1 sho ws that R n d ( n ) y ( n ) ( y x ) = 8><>: 0 ; if x lies outside n 1 ; if x lies inside n ; (B.6) where n is a closed region of in tegration. F urthermore that if g is w ell-b eha v ed in n ; then R n d ( n ) y g ( y ) @ @ y ( n ) ( y x ) = @ g ( y ) @ y y = x (B.7) after an in tegration b y parts. If n is all of space, then the in tegral in ( B.6 ) is alw a ys 1 and the function g in ( B.7 ) need only v anish at innit y Notice that the dimensions of the -function m ust cancel the dimensions of the dieren tial d ( n ) y to giv e a dimensionless result. Hence ( n ) has dimensions of (length) n : There are a n um b er of additional prop erties of the -function whic h will not b e elab orated on here. The in terested reader is referred to [ 7 ] for a detailed discussion.

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118 B.2 The r 2 Op erator Consider the P oisson equation from electro dynamics r 2 = 4 ; (B.8) where is the scalar p oten tial and is the c harge densit y The corresp onding Green's function satises the equation r 2 G ( x ; x 0 ) = ( x x 0 ) ; (B.9) where the dimensionalit y of the -function has b een omitted. By going to the F ourier space ( B.9 ) is diagonalized and b ecomes ( i k ) 2 ~ G = exp ( i k x 0 ) : By another F ourier in v ersion G ( x ; x 0 ) = R V d 3 k (2 ) 3 e i k ( x x 0 ) k 2 = 1 4 j x x 0 j : (B.10) With ( B.10 ) the P oisson equation ( B.8 ) is recast as the in tegral equation ( x ; t ) = R V d 3 x 0 [ 4 ( x 0 ; t )] G ( x 0 ; x ) = R V d 3 x 0 ( x 0 ; t ) j x x 0 j : (B.11) T ogether with the homogeneous solutions of the Laplace equation r 2 = 0 ; the total solution is obtained. B.3 The @ 2 Op erator Consider the w a v e equation from electro dynamics @ 2 = = r 2 c 2 = 4 ; (B.12) where is the scalar p oten tial and is the c harge densit y as b efore. The corresp onding Green's function satises h r 2 1 c 2 @ 2 @ t 2 i G ( x ; t ; x 0 ; t 0 ) = ( x x 0 ) ( t t 0 ) : (B.13)

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119 The solution of ( B.13 ) will require the F ourier in v ersion in b oth x and t to obtain [ k 2 2 =c 2 ] ~~ G = exp ( i [ k x 0 t 0 ]) : Division b y [ k 2 2 =c 2 ] follo w ed b y another F ourier in v ersion in b oth k and will giv e the desired result. Rather than p erforming this task in one step, it is b enecial to split the eort in half. First consider the F ourier in v erse of ( B.12 ) in x : That is ~ + ( c k ) 2 ~ = 4 c 2 ~ ; (B.14) with Green's function satisfying h ( d 2 =dt 2 ) + ( c k ) 2 i D k ( t; t 0 ) = ( t t 0 ) : (B.15) F ourier in v ersion in t results in [ 2 c 2 k 2 ] ~ D k = exp ( i! t 0 ) from whic h the retarded Green's function b ecomes D (+) k ( t; t 0 ) = R 11 d! 2 e i! ( t t 0 ) 2 c 2 k 2 = ( t t 0 ) sin ck ( t t 0 ) ck ; (B.16) where the in tegration w as p erformed in lo w er half complex plane. Similarly the adv anced solution is D ( ) k ( t; t 0 ) = ( t 0 t ) sin ck ( t t 0 ) =ck b y in tegrating in the upp er half plane. The solution to ( B.13 ) is more clear, no w that this rst step has b een accomplished. It has already b een sho wn that [ k 2 2 =c 2 ] ~~ G = exp ( i [ k x 0 t 0 ]) m ust b e F ourier in v erted t wice in order to obtain G: The resulting Green's function is G ( x ; t ; x 0 ; t 0 ) = c 2 R V d 3 k (2 ) 3 e i k ( x x 0 ) R 11 d! 2 e i! ( t t 0 ) 2 c 2 k 2 = c ( t t 0 ) 2 2 j x x 0 j R 10 dk sin ck ( t t 0 ) sin( k j x x 0 j ) = c ( t t 0 ) 4 j x x 0 j h j x x 0 j c ( t t 0 ) j x x 0 j + c ( t t 0 ) i ; (B.17) where the retarded Green's function D (+) in ( B.16 ) w as used in the rst equalit y and a c hange of v ariables w as used in the last equalit y The rst term in ( B.17 ) is

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120 referred to as the r etar de d solution G (+) ( x ; t ; x 0 ; t 0 ) = c ( t t 0 ) 4 j x x 0 j j x x 0 j c ( t t 0 ) (B.18) while the second term is just zero since b oth j x x 0 j and c ( t t 0 ) are p ositiv e. An analogous computation with the adv anced Green's function D ( ) leads to the advanc e d solution G ( ) ( x ; t ; x 0 ; t 0 ) = c ( t 0 t ) 4 j x x 0 j j x x 0 j + c ( t t 0 ) : (B.19) The retarded solution exhibits the causal prop erties of eld propagation (see Figure B{1 ). That is, a disturbance that is observ ed at the p oin t ( x ; t ) in space-time PSfrag replacemen ts r ( t ) ct x past elsewhere future Figure B{1: The tra jectory or w orld line r ( t ) of a massiv e particle mo v es from past to future within the ligh t cone. A massless particle suc h as a photon propagates on the ligh t cone. originated from a p oin t that is a distance j x x 0 j = c ( t t 0 ) a w a y and at a time t 0 = t j x x 0 j =c earlier. The opp osite is true for the adv anced solution. With ( B.18 ) or ( B.19 ) the w a v e equation ( B.12 ) is recast as the in tegral equation ( x ; t ) = R V d 3 x 0 R 11 dt 0 [ 4 ( x 0 ; t )] G ( ) ( x 0 ; t 0 ; x ; t ) = R V d 3 x 0 R 11 dt 0 ( x 0 ; t 0 ) j x x 0 j t 0 h t j x x 0 j c i : (B.20)

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121 T ogether with the homogeneous solutions of the w a v e equation = 0 ; the total solution is obtained.

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APPENDIX C THE TRANSVERSE PR OJECTION OF A ( x ; t ) It can b e seen in b oth ( 3.56 ) and ( 3.59 ) that the full v ector p oten tial generated b y a c harge q mo ving with v elo cit y v also p oin ts in the direction of v : When w orking in the Coulom b gauge it is not the full v ector p oten tial that is needed but the transv erse pro jection thereof. The tr ansverse ve ctor p otential A T do es not ro w in the direction p erp endicular to v ; but rather the direction p erp endicular to the w a v ev ector k as seen in Figure C{1 The true meaning of transv erse and longitudinal PSfrag replacemen ts r ( t ) v k v ? k ( k v ) =k 2 v k k ( k v ) =k 2 Figure C{1: Since ~ A = ~ h v ; the transv erse v ector p oten tial ~ A ? = [ v k ( k v ) =k 2 ] ~ h and the longitudinal v ector p oten tial ~ A k = [ k ( k v ) =k 2 ] ~ h; where ~ h is a scalar function. is easily visualized b y going to the F ourier space. There i k ~ A ? = 0 and i k ~ A k = 0 ; where ~ A ? is the F ourier in v erse of A T and ~ A k is the F ourier in v erse of the longitudinal A L : The transv erse and longitudinal pro jections satisfy A T + A L = A : Eac h comp onen t A iT of the A T can b e obtained b y con traction of the transv erse 122

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123 pro jection tensor T ij with the v elo cit y v j That is A iT ( x ; t ) = T ij ( x ; t ) v j (C.1) where T ij ( x ; t ) is related to the transv erse pro jection op erator P ij T = ij @ i @ j = r 2 according to P ij T h ( x ; t ) v j | {z } A j ( x ;t ) = h ij @ i @ j r 2 i h ( x ; t ) v j = [ @ i @ j r 2 ij ] n h ( x ; t ) r 2 o | {z } g ( x ;t ) v j = T ij ( x ; t ) v j : (C.2) Note that P ik T = P ij T P j k T and P y = P : In F ourier space T ij is of the form ~ T ij ( k ; t ) = [ k i k j k 2 ij ] ~ g ( k ; t ) = [3 k i k j k 2 ij ] ~ g ( k ; t ) | {z } ~ Q ij ( k ;t ) + [ 2 k i k j ] ~ g ( k ; t ) | {z } ~ M ij ( k ;t ) (C.3) where ~ g distinguishes b et w een the F ourier in v erses of A (0) ; A (1) ; or A (2) : These v ector p oten tials are the pieces whic h mak e up the full p oten tial A = ( t t 1 )( t 2 t )[ A (0) + A (1) ] + ( t t 2 )[ A (1) A (2) ] (C.4) generated b y the curren t ~ J = q v ( t t 1 )( t 2 t ) exp ( i k [ r + v t ] k 2 = 8 ` 2 ) : The rst instan taneous or Coulom b-lik e piece is A (0) ( x ; t ) = v c (0) ( x ; t ) = q v c erf p 2 ` j x ( r + v t ) j j x ( r + v t ) j (C.5) while the remaining three pairs of radiating terms are of the form A ( k ) ( x ; t ) = v c ( k ) ( x ; t ) = q v 2 c n erf p 2 ` [ c ( t t k ) j x ( r + v t k ) j ] j x ( r + v t k ) j erf p 2 ` [ c ( t t k ) + j x ( r + v t k ) j ] j x ( r + v t k ) j o (C.6)

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124 for k = 1 ; 2 : The rst term in ( C.3 ) resem bles the tr ac eless quadrup ole momen t tensor and is lab eled ~ Q ij ; while the second term ~ M ij is the remainder. It will b e seen that the ~ M ij tensor can b e completely determined from ~ Q ij : Solving for eac h comp onen t of T ij rst and then con tracting with the v elo cit y eliminates the F ourier in v ersion of ~ A ? ( k ; t ) = [ k ( k v ) k 2 v ] ~ g ( k ; t ) (C.7) in fa v or of the angularly inferior T ij ( x ; t ) = R V d 3 k (2 ) 3 [ k i k j k 2 ij ] ~ g ( k ; t ) e i k x ; (C.8) follo w ed b y a simple m ultiplication of v j : In matrix notation the transv erse v ector p oten tial is just A T = Tv = [ Q + M ] v : (C.9) Lo oking only at ~ Q ij (although the same is true for b oth ~ M ij and ~ T ij ), it can b e sho wn that while ~ Q ij 6 = 0 for all i and j; its F ourier in v erse Q ij is diagonal in a certain frame. F or example (for A (0) ) ~ Q (0)12 ( k ; t ) = 3 k x k y ~ g (0) C ( k ; t ) = 3 k x k y 4 q ck 4 e i k ( r + v t ) k 2 = 8 ` 2 6 = 0 ; (C.10) while Q (0)12 ( x ; t ) = R V d 3 k (2 ) 3 3 k x k y 4 q ck 4 e i k [ x ( r + v t )] k 2 = 8 ` 2 = 0 (C.11) in the b o osted frame of origin r + v t where x ( r + v t ) is rotated ab out the origin to lie along the ^ e z -axis. As exp ected, if x ( r + v t ) is placed in general along the constan t v ector ^ n 0 = ^ e x sin 0 cos 0 + ^ e y sin 0 sin 0 + ^ e z cos 0 ; then the angular part of Q (0)12 b ecomes R d n sin cos sin sin e ik j x ( r + v t ) j [cos cos 0 +sin sin 0 cos ( 0 )] 6 = 0 : (C.12)

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125 C.1 T ensor Calculus It will b e seen that the transv erse pro jection tensor T ij in ( C.8 ) is diagonal in the b o osted frame of origin r + v t where the general v ector x ( r + v t ) x is rotated ab out the origin to the new v ector x 0 = r (0 ; 0 ; 1) whic h lies along the ^ e z -axis. This spatial rotation is p erformed via the rotation matrix k l (n) = 0BBBB@ cos cos cos sin sin sin cos 0 sin cos sin sin cos 1CCCCA ; (C.13) where x 0 k = k l x l or in matrix notation x 0 = x : With k l ; the transv erse pro jection tensor in ( C.8 ) ma y b e rotated to the diagonal frame b y T k l ( x ) T 0 k l ( x 0 ) = @ x 0 k @ x i @ x 0 l @ x j T ij ( 1 x 0 ) = k i l j T ij ( x ) (C.14) where k i l j T ij = k i T ij [ 1 ] l j or T 0 = T 1 : The explicit time-dep endence whic h is unaected b y the rotation has b een dropp ed for brevit y Applying this similarit y transformation to T ij results in T 0 k l ( x 0 ) = k i T ij ( x )[ 1 ] l j = R V d 3 k (2 ) 3 e i k x k i ~ T ij ( k )[ 1 ] l j = R V d 3 k 0 (2 ) 3 e i k 0 x 0 ~ T 0 k l ( k 0 ) ; (C.15) where d 3 k 0 = det d 3 k = d 3 k and k 0 x 0 = k x : In this frame the three diagonal elemen ts of T 0 k l are kno wn. They are 0BBBB@ T 0 11 ( x 0 ) 0 0 0 T 0 22 ( x 0 ) 0 0 0 T 0 33 ( x 0 ) 1CCCCA = R V d 3 k 0 (2 ) 3 e i k 0 x 0 0BBBB@ ~ T 0 11 ( k 0 ) ~ T 0 12 ( k 0 ) ~ T 0 13 ( k 0 ) ~ T 0 21 ( k 0 ) ~ T 0 22 ( k 0 ) ~ T 0 23 ( k 0 ) ~ T 0 31 ( k 0 ) ~ T 0 32 ( k 0 ) ~ T 0 33 ( k 0 ) 1CCCCA : (C.16) Eac h of the elemen ts T 0 ii ( x 0 ) will b e computed in the diagonal frame. Ho wev er the ph ysics is not correct un til x 0 is rotated bac k to the general p osition

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126 x = r (sin cos ; sin sin ; cos ) : The rev erse rotation is obtained b y in v ersion of the transformation ( C.14 ). The resulting transv erse pro jection tensor b ecomes T k l ( x ) = 0BBBB@ cos 2 cos 2 cos 2 sin cos sin cos cos cos 2 sin cos cos 2 sin 2 sin cos sin sin cos cos sin cos sin sin 2 1CCCCA T 0 11 ( x 0 ) + 0BBBB@ sin 2 sin cos 0 sin cos cos 2 0 0 0 0 1CCCCA T 0 22 ( x 0 ) + 0BBBB@ sin 2 cos 2 sin 2 sin cos sin cos cos sin 2 sin cos sin 2 sin 2 sin cos sin sin cos cos sin cos sin cos 2 1CCCCA T 0 33 ( x 0 ) (C.17) whic h ma y b e written in matrix form as T = 1 T 0 : The transv erse v ector p otential A kT ( x ) is no w obtained b y the simple con traction T k l ( x ) v l : Note that the T 0 k k ( x 0 ) app earing in ( C.17 ) are scalar functions that only dep end up on the norm of x 0 : A prop er treatmen t of the tensor calculus rev eals that r 0 A 0T ( x 0 ; t ) = r A T ( x ; t ) : Comp onen t wise that is @ 0 k T 0 k l ( x 0 ; t ) v 0 l = @ i T ij ( x ; t ) v j : (C.18) It m ust still b e v eried that A T is div ergenceless in either frame. It is sucien t to sho w that @ i T ij v j = 0 : This result is most easily sho wn b y w orking in the ( r ; ; ) basis with unit v ectors ^ e x = ^ e r sin cos + ^ e cos cos ^ e sin ^ e y = ^ e r sin sin + ^ e cos sin + ^ e cos ^ e z = ^ e r cos ^ e sin : (C.19)

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127 In these spherical co ordinates the the v elo cit y b ecomes v = 0BBBB@ v x v y v z 1CCCCA = v 1 0BBBB@ sin cos cos cos sin 1CCCCA + v 2 0BBBB@ sin sin cos sin cos 1CCCCA + v 3 0BBBB@ cos sin 0 1CCCCA (C.20) and the transv erse v ector p oten tial is just A T = v 1 0BBBB@ T 33 sin cos T 11 cos cos T 22 sin 1CCCCA + v 2 0BBBB@ T 33 sin sin T 11 cos sin T 22 cos 1CCCCA + v 3 0BBBB@ T 33 cos T 11 sin 0 1CCCCA : (C.21) It is not dicult to v erify that r A T = 0 using the spherical div ergence. T o summarize the w ork so far, it w as stated that a frame exists where the transv erse pro jection tensor T ij is diagonal. A spatial rotation w as p erformed to go to that diagonal frame. The tensor w as then rotated bac k to the arbitrary frame. There the matrix elemen ts of the general T ij in v olv e the diagonal elemen ts T 0 ii as seen in ( C.21 ). In the follo wing section the three T 0 ii terms corresp onding to the three terms in the v ector p oten tial A = ( t t 1 )( t 2 t )[ A (0) + A (1) ] + ( t t 2 )[ A (1) A (2) ] will b e computed. C.2 T 0 k k ( x 0 ; t ) In tegrals In ( C.3 ) and ( C.9 ) it w as sho wn that the transv erse v ector p oten tial ma y b e obtained b y con traction of the transv erse pro jection tensor T = Q + M with the v elo cit y That is A iT ( x ; t ) = R V d 3 k (2 ) 3 [ ~ Q ij ( k ; t ) + ~ M ij ( k ; t )] e i k x v j = R V d 3 k (2 ) 3 [3 k i k j k 2 ij ] ~ g ( k ; t ) e i k x v j + R V d 3 k (2 ) 3 [ 2 k i k j ] ~ g ( k ; t ) e i k x v j ; (C.22)

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128 where it w as noticed that ~ Q ij is analogous to the traceless quadrup ole momen t tensor from electro dynamics [ 9 ]. In the diagonal frame ~ Q 0 ij lo oks lik e ~ Q 0 ij ( x 0 ; t ) = 0BBBB@ ~ Q 0 11 ( x 0 ; t ) 0 0 0 ~ Q 0 22 ( x 0 ; t ) 0 0 0 ~ Q 0 33 ( x 0 ; t ) 1CCCCA (C.23) with ~ Q 0 11 + ~ Q 0 22 + ~ Q 0 33 = 0 : The elemen ts of the M tensor can all b e found from Q : Recall from ( 3.58 )-( 3.59 ) or ( C.4 )-( C.6 ) that the piecewise v ector p oten tial A = ( t t 1 )( t 2 t )[ A (0) + A (1) ] + ( t t 2 )[ A (1) A (2) ] (C.24) is made up of a Coulom b-lik e piece A (0) ( x ; t ) = v c (0) ( x ; t ) = q v c erf p 2 ` j x ( r + v t ) j j x ( r + v t ) j (C.25) and three pairs of radiating terms A ( k ) ( x ; t ) = v c ( k ) ( x ; t ) = q v 2 c n erf p 2 ` [ c ( t t k ) j x ( r + v t k ) j ] j x ( r + v t k ) j erf p 2 ` [ c ( t t k ) + j x ( r + v t k ) j ] j x ( r + v t k ) j o (C.26) for k = 1 ; 2 : F or eac h of these pieces there is a corresp onding transv erse pro jection tensor. F or example, when t 1 t t 2 there is a T (0) = Q (0) + M (0) asso ciated with A (0) and a T (1) = Q (1) + M (1) asso ciated with A (1) : Eac h of these tensors in v olv e a F ourier in v ersion. The resulting in tegrals are computed b elo w.

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129 C.2.1 Inside Step When t 1 t t 2 ; the v ector p oten tial A in = A (0) + A (1) : The traceless part of the transv erse pro jection tensor is Q ij ( x ; t ) = R V d 3 k (2 ) 3 [3 k i k j k 2 ij ] f ~ g (0) C ( k ; t ) + ~ g (1) R ( k ; t ) g e i k x = R V d 3 k (2 ) 3 [3 k i k j k 2 ij ] 8>><>>: 4 q ck 4 e i k ( r + v t ) k 2 = 8 ` 2 + 4 q ck 4 e i k ( r + v t 1 ) k 2 = 8 ` 2 cos ck ( t t 1 ) 9>>=>>; e i k x (C.27) and M ij ( x ; t ) = R V d 3 k (2 ) 3 [ 2 k i k j ] f ~ g (0) C ( k ; t ) + ~ g (1) R ( k ; t ) g e i k x (C.28) where k 2 ~ g (0) C v is the F ourier in v erse of A (0) and k 2 ~ g ( k ) R v is the F ourier in v erse of A ( k ) : Higher order terms in v =c are omitted from ( C.27 ). T aking ~ g in ( C.22 ) as ~ g (0) C ; the Q (0)ij tensor b ecomes Q (0)ij = R V d 3 k (2 ) 3 [3 k i k j k 2 ij ] 4 q ck 4 e i k [ x ( r + v t )] k 2 = 8 ` 2 : (C.29) Using cylindrical symmetry it is found that all o-diagonal elemen ts of Q (0)ij are zero and Q (0)11 = Q (0)22 when the v ector x ( r + v t ) is rotated to the frame where it lies along the z -axis. But since Q (0)ij is traceless, Q (0)33 = 2 Q (0)11 : Th us, solving for Q (0)33 determines Q (0) en tirely That is Q (0)33 = R V d 3 k (2 ) 3 [3 k 2 z k 2 ] 4 q ck 4 e i k [ x ( r + v t )] k 2 = 8 ` 2 = 3 q c R 10 dz cos ( j x ( r + v t ) j z ) e z 2 = 16 ` 2 W 1 ; 1 = 2 z 2 8 ` 2 + (0) c = 2 q ` p 2 c 2 F 2 1 2 ; 3 2 ; 1 2 ; 5 2 ; 2 ` 2 j x ( r + v t ) j 2 + (0) c = 3 q 8 ` 2 c erf p 2 ` j x ( r + v t ) j j x ( r + v t ) j 3 + 3 q p 8 `c e 2 ` 2 [ x ( r + v t )] 2 j x ( r + v t ) j 2 + (0) c ; (C.30)

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130 where the primes whic h denote this diagonal frame ha v e b een temp orarily omitted and where Gradsh teyn and Ryzhik [ 87 ] w as used. Notice that Q (0)33 not only determines Q (0)11 and Q (0)22 ; it also determines M (0) 11 ; M (0) 22 ; and M (0) 33 : T aking ~ M (0) ij as ~ M (0) ij = [ 2 k i k j ] 4 q ck 4 e i k ( r + v t ) k 2 = 8 ` 2 (C.31) it is found that M (0) 11 = M (0) 22 = [ cQ (0)33 + 2 (0) ] = 3 c and M (0) 33 = 2[ cQ (0)33 (0) ] = 3 c with M (0) ij = 0 for i 6 = j: With this, the v ector p oten tial transv erse to A (0) b ecomes A (0)T = [ Q (0) + M (0) ] v : (C.32) T aking ~ g in ( C.22 ) as ~ g (1) R ; the Q (1)ij tensor b ecomes Q (1)ij = R V d 3 k (2 ) 3 [3 k i k j k 2 ij ] 4 q ck 4 e i k [ x ( r + v t 1 )] k 2 = 8 ` 2 cos ck ( t t 1 ) : (C.33) Using spherical symmetry it is again found that all o-diagonal elemen ts of Q (1)ij are zero and Q (1)11 = Q (1)22 when the v ector x ( r + v t 1 ) is c hosen to lie along the z -axis. Again since Q (1)ij is traceless, Q (1)33 = 2 Q (1)11 : Th us, solving for Q (1)33 determines Q (1) en tirely That is Q (1)33 = R V d 3 k (2 ) 3 [3 k 2 z k 2 ] 4 q ck 4 e i k [ x ( r + v t 1 )] k 2 = 8 ` 2 cos ck ( t t 1 ) = 12 q ` p 2 c e 2 ` 2 c 2 ( t t 1 ) 2 R 10 du u 2 e 2 ` 2 [ x ( r + v t 1 )] 2 u 2 cosh 4 ` 2 c ( t t 1 ) j x ( r + v t 1 ) j u + (1) c = 3 q p 8 `c n [ c ( t t 1 ) j x ( r + v t 1 ) j ] e 2 ` 2 [ c ( t t 1 )+ j x ( r + v t 1 ) j ] 2 j x ( r + v t 1 ) j 3 [ c ( t t 1 ) + j x ( r + v t 1 ) j ] e 2 ` 2 [ c ( t t 1 ) j x ( r + v t 1 ) j ] 2 j x ( r + v t 1 ) j 3 o + 3 q 8 ` 2 c 1 + 4 ` 2 c 2 ( t t 1 ) 2 j x ( r + v t 1 ) j 3 n erf p 2 ` [ c ( t t 1 ) + j x ( r + v t 1 ) j ] erf p 2 ` [ c ( t t 1 ) j x ( r + v t 1 ) j ] o + (1) c ; (C.34)

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131 where Gradsh teyn and Ryzhik [ 87 ] w as used. Similarly Q (1)33 determines M (1) 11 ; M (1) 22 ; and M (1) 33 : T aking ~ M (1) ij as ~ M (1) ij = [ 2 k i k j ] 4 q ck 4 e i k ( r + v t 1 ) k 2 = 8 ` 2 cos ck ( t t 1 ) (C.35) it is found that M (1) 11 = M (1) 22 = [ cQ (1)33 + 2 (1) ] = 3 c and M (1) 33 = 2[ cQ (1)33 (1) ] = 3 c with M (1) ij = 0 for i 6 = j: With this, the v ector p oten tial transv erse to A (1) b ecomes A (1)T = [ Q (1) + M (1) ] v (C.36) so that within the step A Tin = A (0)T + A (1)T : C.2.2 Outside Step When t > t 2 ; the v ector p oten tial A out = A (1) A (2) : The traceless part of the transv erse pro jection tensor is Q ij ( x ; t ) = R V d 3 k (2 ) 3 [3 k i k j k 2 ij ] f ~ g (1) R ( k ; t ) ~ g (2) R ( k ; t ) g e i k x = R V d 3 k (2 ) 3 [3 k i k j k 2 ij ] 8>><>>: 4 q ck 4 e i k ( r + v t 1 ) k 2 = 8 ` 2 cos ck ( t t 1 ) 4 q ck 4 e i k ( r + v t 2 ) k 2 = 8 ` 2 cos ck ( t t 2 ) 9>>=>>; e i k x (C.37) and M ij ( x ; t ) = R V d 3 k (2 ) 3 [ 2 k i k j ] f ~ g (1) R ( k ; t ) ~ g (2) R ( k ; t ) g e i k x ; (C.38) where k 2 ~ g ( k ) R v is the F ourier in v erse of A ( k ) :

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132 T aking ~ g in ( C.22 ) as ~ g (1) R ; the z z -comp onen t of the tensor Q (1)ij is Q (1)33 = 3 q p 8 `c n [ c ( t t 1 ) j x ( r + v t 1 ) j ] e 2 ` 2 [ c ( t t 1 )+ j x ( r + v t 1 ) j ] 2 j x ( r + v t 1 ) j 3 [ c ( t t 1 ) + j x ( r + v t 1 ) j ] e 2 ` 2 [ c ( t t 1 ) j x ( r + v t 1 ) j ] 2 j x ( r + v t 1 ) j 3 o + 3 q 8 ` 2 c 1 + 4 ` 2 c 2 ( t t 1 ) 2 j x ( r + v t 1 ) j 3 n erf p 2 ` [ c ( t t 1 ) + j x ( r + v t 1 ) j ] erf p 2 ` [ c ( t t 1 ) j x ( r + v t 1 ) j ] o + (1) c (C.39) and as b efore M (1) 11 = M (1) 22 = [ cQ (1)33 + 2 (1) ] = 3 c and M (1) 33 = 2[ cQ (1)33 (1) ] = 3 c with M (1) ij = 0 for i 6 = j: Lastly taking ~ g in ( C.22 ) as ~ g (2) R ; the z z -comp onen t of Q (2)ij is found to b e Q (2)33 = 3 q p 8 `c n [ c ( t t 2 ) j x ( r + v t 2 ) j ] e 2 ` 2 [ c ( t t 2 )+ j x ( r + v t 2 ) j ] 2 j x ( r + v t 2 ) j 3 [ c ( t t 2 ) + j x ( r + v t 2 ) j ] e 2 ` 2 [ c ( t t 2 ) j x ( r + v t 2 ) j ] 2 j x ( r + v t 2 ) j 3 o + 3 q 8 ` 2 c 1 + 4 ` 2 c 2 ( t t 2 ) 2 j x ( r + v t 2 ) j 3 n erf p 2 ` [ c ( t t 2 ) + j x ( r + v t 2 ) j ] erf p 2 ` [ c ( t t 2 ) j x ( r + v t 2 ) j ] o + (2) c : (C.40) T aking ~ M (2) ij as ~ M (2) ij = [ 2 k i k j ] 4 q ck 4 e i k ( r + v t 1 ) k 2 = 8 ` 2 cos ck ( t t 1 ) (C.41) it is found that M (2) 11 = M (2) 22 = [ cQ (2)33 + 2 (2) ] = 3 c and M (2) 33 = 2[ cQ (2)33 (2) ] = 3 c with M (2) ij = 0 for i 6 = j: With this, the v ector p oten tial transv erse to A (2) b ecomes A (2)T = [ Q (2) + M (2) ] v (C.42) so that outside the step A Tout = A (1)T A (2)T :

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133 C.3 Building A T ( x ; t ) With the result of equations ( C.30 ), ( C.34 ), ( C.39 ), and ( C.40 ), the transv erse v ector p oten tial can b e determined b y con traction of Q ij + M ij with the v elo cit y v j : As a result A T b ecomes A T = [ Q + M ] v = ( t t 1 )( t 2 t )[ A (0)T + A (1)T ] + ( t t 2 )[ A (1)T A (2)T ] : (C.43) These results are presen ted in [ 88 ].

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BIOGRAPHICAL SKETCH Da vid John Masiello w as b orn on Octob er 8th, 1977, in Pro vidence, Rho de Island and w as the only c hild of John Alfred Masiello and Norma Jean Masiello. Although his immediate family w as small, Da vid w as part of a large Italian-American family that gathered religiously ev ery Sunda y for dinner. Da vid's man y cousins w ere more lik e brothers and sisters. T ogether they sp en t endless da ys catc hing butterries, building tree forts, and practicing their art w ork under the sup ervision of their grandfather. In searc h of w arm sunshine and blue skies, Da vid's paren ts decided to lea v e New England. They mo v ed to sunn y Florida just in time for Da vid to b egin high sc ho ol. While in high sc ho ol Da vid b ecame in terested in c hemistry and biology and en tered the Univ ersit y of Florida in 1995 with the in ten tions of pursuing a career in medicine. These in ten tions quic kly c hanged as Da vid found that his deep er questions could not b e answ ered b y these disciplines. In Ma y of 1999, Da vid receiv ed a B.S. degree in mathematics from the Univ ersit y of Florida. During his undergraduate career, Da vid b ecame in terested in the applications of mathematics in the ph ysical sciences. This in terest led him to carry out researc h in optical ph ysics o v er three univ ersit y campuses w orldwide. Alw a ys striving for a deep er more fundamen tal understanding of Nature, Da vid decided to sta y at the Univ ersit y of Florida to earn a Ph.D. under the advisemen t of Prof. Yngv e Ohrn and Dr. Erik Deumens at the Quan tum Theory Pro ject. During his third y ear of graduate sc ho ol Da vid married his college sw eetheart, Kathryn Allida Masiello. 142


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ON THE CANONICAL FORMULATION OF ELECTRODYNAMICS AND
WAVE MECHANICS















By

DAVID JOHN MASIELLO


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2004



















For Katie.















ACKNOWLEDGMENTS

Since August of 1999, I have had the privilege of conducting my Ph.D. research

in the group of Prof. Yngve Ohrn and Dr. Erik Deumens at the University of

Florida's Quantum Theory Project. During my time in their group I learned a great

deal on the theory of dynamics, in particular, the Hamiltonian approach to dynam-

ics and its applications in electrodynamics and atomic and molecular collisions. I

also learned a new appreciation for scientific computing, of which I was previously

ignorant. Most importantly, Prof. Ohrn and Dr. Deumens taught me how to think

through a 1ir, -i .,1 problem, sort out its underlying dynamical equations, and solve

them in a mathematically well-defined manner. I especially want to thank Dr. Erik

Deumens, with whom I worked most closely during my Ph.D. research. Erik had

a vision when I began my graduate studies and has promoted my work since then

to successfully realize it. Along the way, he challenged my creative, mathematical,

and ].li,--i, .1l intuitions and imparted on me a love for theoretical ]li-,-i. Erik has

always taken time to listen to and carefully answer my questions and has always

respected my ideas. I thank him for being such an excellent mentor to me.

My understanding of 1.1r,-i. has also been broadened by many others. Firstly, I

would like to thank Dr. Remigio Cabrera-Trujillo, who was a post doctoral associate

in the Ohrn-Deumens group, for his guidance especially during my first few years.

He has been a great source for advice on many topics from the details of quantum

scattering theory to simple computer problems like clearing printer jams. I have

joked on many occasions that he was my personal postdoc because he was always

so willing to help when I had questions. I would also like to thank the past and

present members of my research group, in particular, Dr. Anatol Blass, Dr. Mauricio









Coutinho Neto, Mr. Ben Killian, and Mr. Virg Fermo. In addition, I would like

to thank my officemates with whom I have spent almost five years. I thank Ms.

Ariana Beste, Mr. Igor Schweigert, and Mr. Tom Henderson for their friendship

and camaraderie. I have especially benefited from many conversations with Tom

Henderson on aspects of quantum mechanics, quantum field theory, and classical

electrodynamics.

Several other faculty and staff at the Quantum Theory Project, and the Depart-

ments of Chemistry, Physics, and Mathematics at the University of Florida have also

encouraged and promoted my Ph.D. research. At the Quantum Theory Project, I

thank Prof. Jeff Krause for taking sincere interest in my research and always finding

time to listen to me and provide guidance. I have taught with Jeff on a few occasions

and have known him to be a great teacher as well as mentor. I thank Prof. Henk

Monkhorst for his kindness and good humor. I will especially miss all of the IATEX

battles that we have fought over the past several years. In addition, I would like to

thank Dr. Ajith Perera for his friendship and patience. I thank the staff, especially

Ms. Judy Parker and Ms. Coralu Clements, for keeping all of the administrative

aspects of my graduate studies running smoothly. I would also like to thank the

custodians Sandra and Rhonda who have been so friendly to me and who keep the

Quantum Theory Project impeccably clean. In the Department of Chemistry, I

would like to thank the late Prof. Carl Stoufer, who was my undergraduate advisor

during my first year, for his friendship, wisdom, and advice. Throughout my entire

undergraduate career we would meet a few times per year to catch up over coffee and

donuts. It was due to Carl's support that I was given the opportunity to study at

the Quantum Theory Project. In the Department of Physics, I would like to thank

Prof. Richard Woodard, from whom I learned quantum field theory. Richard is very

passionate about ]li-, -i. and is perhaps the best teacher that I have known. From

him I gained a deeper understanding of perturbation theory and its applications in









quantum electrodynamics. In the Department of Mathematics, I would like to thank

Prof. Scott McCullough, who was effectively my undergraduate advisor. While I

was an undergraduate student of Scott's, he imparted to me a deep appreciation for

mathematics and a particular interest in analysis. Scott was an excellent teacher

and mentor, and under his guidance, my undergraduate research was awarded by

the College of Liberal Arts and Sciences.

Outside of the University of Florida, many others have contributed to my scien-

tific career. At the University of Central Florida's Center for Research and Education

in Optics and Lasers, I would like to thank Prof. Leonid Glebov, Prof. Kathleen

Richardson, and Prof. Boris Zel'dovich for first introducing me to the world of quan-

tum .l, 1-i' In particular, Prof. Glebov and Prof. Richardson greatly stimulated

and encouraged my interests. With their recommendation, I received a fellowship

to study at the University of Bordeaux's Department of Physics and Centre de

Physique Mol6culaire Optique et Herzienne. While in Bordeaux, France, I had the

pleasure of working in the research group of Prof. Laurent Sarger. I wish to thank

Prof. Sarger as well as his colleagues for their hospitality during my time in France

and for introducing me to the field of atomic and molecular 1ir, -i. -. which is the

setting for this dissertation.

Lastly, I would like to thank my family. My mother and father have always

provided unconditional love, support, and guidance to me. They have encouraged

my inquisitiveness of Nature and have promoted my education from kindergarten

to Ph.D. I thank my inlaws for their love and support and for providing a home

away from home while in graduate school. In conclusion, I would like to thank my

wonderful wife Katie for her encouragement, companionship, and unending love.
















TABLE OF CONTENTS


page


ACKNOW LEDGMENTS .............................

LIST OF FIGURES ................................

A B ST R A C T . . . . . . . . .

CHAPTER

1 INTRODUCTION ..............................

1.1 Physical M otivation .. .. .. ... .. .. .. .. ... .. .. .
1.2 Historical and Mathematical Background .............
1.2.1 Gauge Symmetry of Electrodynamics ............
1.2.2 Gauge Symmetry of Electrodynamics and Wave Mechanics
1.3 Approaches to the Solution of the Maxwell-Schr6dinger Equations
1.4 Canonical Formulation of the Maxwell-Schr6dinger Equations .
1.5 Form at of Dissertation .. ...................
1.6 Notation and Units . .. ... .. .. .. .. ... .. .. .

2 THE DYNAM ICS .. . ........................


2.1 Lagrangian Formalism . ............
2.1.1 Hamilton's Principle . . .....
2.1.2 Example: The Harmonic Oscillator in (qk, ) .
2.1.3 Geometry of TQ . ...........
2.2 Hamiltonian Formalism . . .
2.2.1 Example: The Harmonic Oscillator in (qa, a)
2.2.2 Symplectic Structure and Poisson Brackets .
2.2.3 Geometry of T*Q . ..........

3 ELECTRODYNAMICS AND QUANTUM MECHANICS .


. . 18
. . 19
. . 21
. . 22
..24
. . 25
. . 26
. . 27

. . 29


3.1 Quantum Mechanics in the Presence of an Electromagnetic Field 29
3.1.1 Time-Dependent Perturbation Theory . . .... 30
3.1.2 Fermi Golden Rule . . . 33
3.1.3 Absorption of Electromagnetic Radiation by an Atom .. 34
3.1.4 Quantum Electrodynamics in Brief . . ..... 36
3.2 Classical Electrodynamics Specified by the Sources p and J .. 40
3.2.1 Electromagnetic Radiation from an Oscillating Source .. 41
3.2.2 Electromagnetic Radiation from a Gaussian Wavepacket .47









4 CANONICAL STRUCTURE ........ ................ 55

4.1 Lagrangian Electrodynamics ......... ........ .... 56
4.1.1 Choosing a Gauge ......... ........ ..... 56
4.1.2 The Lorenz and Coulomb Gauges . . 57
4.2 Hamiltonian Electrodynamics ........... ... .. 59
4.2.1 Hamiltonian Formulation of the Lorenz Gauge ...... 61
4.2.2 Poisson Bracket for Electrodynamics . . . 66
4.3 Hamiltonian Electrodynamics and Wave Mechanics in Complex
Phase Space ...... ....... ... ......... 66
4.4 Hamiltonian Electrodynamics and Wave Mechanics in Real Phase
Space ......... .. ..... ............ 69
4.5 The Coulomb Reference by Canonical Transformation ...... 70
4.5.1 Symplectic Transformation to the Coulomb Reference 71
4.5.2 The Coulomb Reference by Change of Variable ...... 78
4.6 Electron Spin in the Pauli Theory ..... . . 79
4.7 Proton Dynamics ............... ....... .. 81

5 NUMERICAL IMPLEMENTATION .............. .. .. 84

5.1 Maxwell-Schrodinger Theory in a Complex Basis . ... 85
5.2 Maxwell-Schrodinger Theory in a Real Basis . . .... 88
5.2.1 Overview of Computer Program . . ..... 90
5.2.2 Stationary States: s- and p-Waves . . 93
5.2.3 Nonstationary State: Mixture of s- and p-Waves . 93
5.2.4 Free Electrodynamics . . ....... ... 93
5.2.5 Analysis of Solutions in Numerical Basis . ... 95
5.3 Symplectic Transformation to the Coulomb Reference ...... 99
5.3.1 Numerical Implementation ... . . ... 101
5.3.2 Stationary States: s- and p-Waves . . ... 102
5.3.3 Nonstationary State: Mixture of s- and p-Waves . 102
5.3.4 Free Electrodynamics .... . ... 103
5.3.5 Analysis of Solutions in Coulomb Basis . .... 103
5.4 Asymptotic Radiation ................ .. ..103
5.5 Proton Dynamics in a Real Basis ... . . .... 108

6 CONCLUSION .............. . . ... 110

APPENDIX

A GAUGE TRANSFORMATIONS ................ ... ..113

A.1 Gauge Symmetry of Electrodynamics . . . 113
A.2 Gauge Symmetry of Quantum Mechanics . . . 115









B GREEN'S FUNCTIONS .......


B.1 The Dirac 6-Function ............
B.2 The V2 Operator ..............
B.3 The 02 Operator ..............

C THE TRANSVERSE PROJECTION OF A(x, t)

C. 1 Tensor Calculus ...............
C.2 T'kk (x t) Integrals .............
C.2.1 Inside Step ..............
C.2.2 Outside Step .............
C.3 Building AT(x, ) .............

REFERENCES ......................

BIOGRAPHICAL SKETCH ...............















LIST OF FIGURES
Figure page

2-1 The configuration manifold Q 2 is depicted together with the
tangent plane TqkQ at the point qk E Q. ............... 22

3-1 The coefficient 1 of the unscattered plane wave exp(ik x) is analo-
gous to the 1 part of the S-matrix, while the scattering amplitude
fk(Q) which modulates the scattered spherical wave exp(ikr)/r is
analogous to the iT part. .................. .... 38

3-2 In the radiation zone, the observation point x is located far from the
source J. In this case the distance |x x' w r f x'. ..... ..44

3-3 The differential power dP/dQ or radiation pattern corresponding to
an oscillating electric dipole verifies that no radiation is emitted in
the direction of the dipole moment. ................ 46

3-4 The norms of J and A are plotted with different velocities along the
x-axis . . . . . . . .. 48

3-5 The trajectory or world line r(t) of the charge is plotted. ...... ..49

3-6 The bremsstrahlung radiation from a charged gaussian wavepacket
moves out on the smeared light cone with maximum at x ct. 50

3-7 The radiation pattern given by (3.63) shows the characteristic dipole
pattern at lowest order. .................. ..... 53

4-1 A limited but relevant portion of the gauge story in the Lagrangian
formalism is organized in this picture. .. . . ..... 59

4-2 The Hamiltonian formulation of the gauge story is organized in this
picture with respect to the previous Lagrangian formulation. . 65

4-3 Commutivity diagram representing the change of coordinates (q, p) to
(p, q) at both the Lagrangian and equation of motion levels. . 79

5-1 Schematic overview of ENRD computer program. . .... 92

5-2 Phase space contour for the coefficients of the vector potential A and
its momentum H. .................. ...... 94

5-3 Phase space contour for the coefficients of the real-valued Schridinger
field Q and its momentum P. .................. .... 94









5-4 Phase space contour for the coefficients of the scalar potential 4 and
its momentum 0. .................. ......... .. 95

5-5 Real part of the Schrodinger coefficients CM(t) {(rr(t)), where
r](t) is a superposition of s- and p,-waves. . . ..... 97
5-6 Imaginary part of the Schrbdinger coefficients Cm(t) (rlr(t)),
where r](t) is a superposition of s- and p,-waves. . . ... 97

5-7 Probability for the electron to be in a particular basis eigenstate. 98

5-8 Phase space of the Schrodinger coefficients CM(t) (rlq (t)), where
r](t) is a superposition of s- and p,-waves. . . ..... 98

5-9 Real part of the Schr6dinger coefficients CM(t) (r Mr(t)), where
r](t) is a superposition of s- and p,-waves. . . 104

5-10 Imaginary part of the Schrodinger coefficients CM(t) = {rlMr(t),
where r](t) is a superposition of s- and p,-waves. . . ... 104

5-11 Probability for the electron to be in a particular basis eigenstate. 105

5-12 Phase space of the Schridinger coefficients Cm(t) (rMq I(t)), where
r](t) is a superposition of s- and p,-waves. . . 105
5-13 Schematic picture of the local and ..1i- ,,i,.i tic basis proposed for the
description of electromagnetic radiation and electron ionization. 107

B-1 The trajectory or world line r(t) of a massive particle moves from past
to future within the light cone. .................. 120

C-1 Since A hv, the transverse vector potential A [v k(k v)/lk2]J
and the longitudinal vector potential A = [k(k v)/k 2]h, where h
is a scalar function. .................. ...... .. 122















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ON THE CANONICAL FORMULATION OF ELECTRODYNAMICS AND
WAVE MECHANICS

By

David John Masiello

\.,v- 2004

Chair: Nils Yngve Ohrn
AI.., ir Department: Chemistry

The interaction of electromagnetic radiation with atoms or molecules is often

understood when the timescale for the electromagnetic decay of an excited state is

separated by orders of magnitude from the timescale of the excited state's dynamics.

In these cases, the two dynamics may be treated separately and a perturbative Fermi

golden rule analysis is appropriate. However, there do exist situations where the

dynamics of the electromagnetic field and the atomic or molecular system occurs

on the same timescale, e.g., photon-exciton dynamics in conjugated polymers and

atom-photon dynamics in cold atom collisions.

Nonperturbative methods for the solution of the coupled nonlinear Maxwell-

Schridinger differential equations are developed in this dissertation which allow

for the atomic or molecular and electromagnetic dynamics to occur on the same

timescale. These equations have been derived within the Hamiltonian or canonical

formalism. The canonical approach to dynamics, which begins with the Maxwell

and Schridinger Lagrangians together with a Lorenz gauge fixing term, yields a

set of first order Hamilton equations which form a well-posed initial value prob-

lem. That is, their solution is uniquely determined and known in principle once









the initial values for each of the associated dynamical variables are specified. The

equations are also closed since the Schridinger wavefunction is chosen to be the

source for the electromagnetic field and the electromagnetic field reacts back upon

the wavefunction.

In practice, the Maxwell-Schrodinger Lagrangian is represented in a basis of

gaussian functions with different widths and centers. Application of the calculus

of variations leads to a set of Euler-Lagrange equations that, for that choice of

basis, form and represent the coupled first order Maxwell-Schridinger equations.

In the limit of a complete basis these equations are exact and for any finite choice

of basis they provide an approximate system of dynamical equations that can be

integrated in time and made systematically more accurate by enriching the basis.

These equations are numerically implemented for a basis of arbitrary finite rank.

The dynamics of the basis-represented Maxwell-Schridinger system is investigated

for the spinless hydrogen atom interacting with the electromagnetic field.















CHAPTER 1
INTRODUCTION

Chemistry encompasses a broad range of Nature that varies over orders of mag-

nitude in energy from the ultracold nK Bose-Einstein condensation temperatures

[1, 2] to the keV collision energies that produce the Earth's aurorae [3-5]. At the

most fundamental level, the study of chemistry is the study of electrons and nuclei.

The interaction of electrons and nuclei throughout this energy regime is mediated

by the photon which is the quantum of the electromagnetic field. The equations

which govern the dynamics of electrons, nuclei, and photons are therefore the same

equations which govern all of chemistry [6]. They are the Schridinger equation [7, 8]


i = HI (1.1)


and Maxwell's equations [9]

4w E B
V-E 4=4p V xB= -J+-E VB 0 VxE+ 0. (1.2)
c c c

As they stand these equations are uncoupled. The solutions of the Schridinger

equation (1.1) do not a priori influence the solutions of the Maxwell equations

(1.2) and vice versa. The development of analytic and numerical methods for the

solution of the coupled Maxwell-Schridinger equations is the main purpose of this

dissertation. Before delving into the details of these methods a 1.li-,-i. .,1 motivation

as well as a historical and mathematical background is provided.

1.1 Physical Motivation

Many situations of ].li,--i. .,1 interest are described by the system of Maxwell-

Schridinger equations. Often these situations involve electromagnetic processes that

occur on drastically different timescales from that of the matter. An example of such









a situation is the stimulated absorption or emission of electromagnetic radiation

by a molecule. The description of this process by (1.1) and (1.2) accounts for a

theoretical understanding of all of spectroscopy, which has provided an immense

body of chemical knowledge.

However, there do exist situations where the dynamics of the electromagnetic

field and the matter occur on the same timescale. For example, in solid state ]li,--i. -

certain electronic wavepackets exposed to strong magnetic fields in semiconductor

quantum wells are predicted to demonstrate rapid decoherence [10]. The dynamics of

the incident field, the electronic wavepacket, and the phonons that it emits is coupled

and occurs on the same femtosecond timescale. In atomic ]'li, -i. -. the long timescale

for the dynamics of cold and ultracold collisions of atoms in electromagnetic traps

has been observed to exceed lifetimes of excited states, which are on the order of 10-s

s. This means that spontaneous emission can occur during the course of collision and

may significantly alter the atomic collision dynamics [11, 12]. Cold atom phenomena

are also being merged with cavity quantum electrodynamics to realize single atom

lasers [13-15]. The function of these novel devices is based on strong coupling of the

atom to a single mode of the resonant cavity. Lastly, in polymer chemistry, ultrafast

light emission has been detected in certain ladder polymer films following ultrafast

laser excitation [16]. A fundamental understanding of the waveguiding process that

occurs in these polymers is unknown. It is p' ..: -, ;- these situations, where the

electromagnetic and matter l.7;/,,.i. occur on the same timescale and are strongly

coupled, that are the motivation for this dissertation.

1.2 Historical and Mathematical Background

The history of the Maxwell-Schridinger equations dates back to the early twen-

tieth century when the founding fathers of quantum mechanics worked out the the-

oretical details of the interaction of electrodynamics with quantum mechanics [17].

It was realized early on that the electromagnetic coupling to matter was through









the potentials D and A, and not the fields E and B themselves [6, 18, 19]. The

potentials and fields are related by


E = -VQ A/c B V x A (1.3)


which can be confirmed by inspecting the homogeneous Maxwell equations in (1.2).

Unlike in classical theory where the potentials were introduced as a convenient math-

ematical tool, the quantum theory requires the potentials and not the fields. That

is, the potentials are fundamental dynamical variables of the quantum theory but

the fields are not. A concrete demonstration of this fact was presented in 1959 by

Aharonov and Bohm [20].

1.2.1 Gauge Symmetry of Electrodynamics

It was well known from the classical theory of electrodynamics [9] that working

with the potentials leads to a potential form of Maxwell's equations that is more

flexible than that in terms of the fields alone (1.2). In potential form, Maxwell's

equations become

A & 4x
V2A VL -V. A +- --J (1.4a)
-c V c

V2b -47p. (1.4b)
C

The homogeneous Maxwell equations are identically satisfied. These potential equa-

tions enjoy a symmetry that is not present in the field equations (1.2). This sym-

metry is called the gauge symmetry and can be generated by the transformation


A -A' A + VF = F/c, (1.5)


where F is a well-behaved but otherwise arbitrary function called the gauge gener-

ator. Applying this gauge transformation to the potentials in (1.4) leads to exactly

the same set of potential equations. In other words, these equations are invari-

ant under arbitrary gauge transformation or are gauge invariant. They possess the









full gauge symmetry. Notice also that the electric and magnetic fields are gauge

invariant. In fact, it turns out that all pli, -i. .,l observables are gauge invariant.

That electrodynamics possesses gauge symmetry places it in a league of theories

known as gauge theories [21]. These theories include general relativity [22, 23] and

Yang-Mills theory [24-26]. Gauge theories all suffer from an indeterminateness due

to their gauge symmetry. In an effort to deal with this indeterminateness, it is

common to first eliminate the symmetry (usually up to the residual symmetry; see

Chapter 4) by gauge f iii and then work within that particular gauge. That is,

the flexibility implied by the gauge transformation (1.5) allows for the potentials to

satisfy certain constraints. These constraints imply a particular choice of gauge and

gauge generator. Gauge fixing is the act of constraining the potentials to satisfy

a certain constraint throughout space-time. For example, in electrodynamics the

potential equations (1.4)


V2A V A + _47--J

Sv (1.4)V A
V2 + --- 47rp
c
form an ill-posed initial value problem. However, they can be converted to a well-

defined initial value problem by adding an equation of constraint to them. For

example, adding the constraint 4/c + V A = 0 leads to the well-defined Lorenz

gauge equations


V2A AJ V2 I = -47p (1.6)
2 c c2

while adding V A = 0 leads to the well-defined Coulomb gauge equations

A 4x
V2A --JT -4p, (1.7)
c2 c

where JT is the transverse projection of the current J (see Appendix A). There

are many other choices of constraint, each leading to a different gauge. It is always









possible to find a gauge function that will transform an arbitrary set of potentials

to satisfy a particular gauge constraint. The subject of the gauge symmetry of

electrodynamics, which is a subtle but fundamental aspect of this dissertation, is

discussed in detail in Chapter 4. In particular, it will be argued that fixing a par-

ticular gauge, which in turn eliminates the gauge from the theory, is not necessarily

optimal. Rather, it is stressed that the gauge freedom is a fundamental variable of

the theory and has its own dynamics.

1.2.2 Gauge Symmetry of Electrodynamics and Wave Mechanics

Since the gauge symmetry of electrodynamics was well known, it was noticed by

the founding fathers that if quantum mechanics is to be coupled to electrodynamics,

then the Schr6dinger equation (1.1) needs to be gauge invariant as well. The most

simple way of achieving this is to require the Hamiltonian appearing in (1.1) to be

of the form
[P qA/c]2
H + V + q, (1.8)
2m

where P is the quantum mechanical momentum, V is the potential energy, and m

is the mass of the charge q. This is in analogy with the Hamiltonian for a classical

charge in the presence of the electromagnetic field [27, 28]. The coupling scheme

embodied in (1.8) is known as minimal w ,pllu, since it is the simplest possible

gauge invariant coupling imaginable. The gauge symmetry inherent in the combined

:-,-1 I in of Schr6dinger's equation and Maxwell's equations in potential form can be

generated by the transformation


A A' A + VF -+ = F/c = exp(iqF/c)I.

(1.9)

The transformation on the wavefunction is called a local gauge transformation and

differs from the global gauge transformation exp(i0), where 0 is a constant. These









global gauge transformations are irrelevant in quantum mechanics where the wave-

function is indeterminate up to a global phase. Application of the gauge transfor-

mation (1.9) to the Schrodinger equation with Hamiltonian (1.8) and to Maxwell's

equations in potential form leads to exactly the same equations after the transfor-

mation. Therefore, like the potential equations (1.4) by themselves, the system of

Maxwell-Schrodinger equations

[P qA/c]2
i [ = + gA/ +v q+q, (1.10)
2m


V2A 2- c V A +- J (1.11a)
A cc

V2 -47p (1.11b)
c

is invariant under the gauge transformation (1.9). There are several other symme-

tries that are enjoyed by this system of equations. For example, they are invariant

under spatial rotations, nonrelativistic (Galilei) boosts, and time reversal. As a

result, the Maxwell-Schrbdinger equations enjoy charge, momentum, angular mo-

mentum, and energy conservation. That each continuous symmetry gives rise to

an associated conservation law was proven by Emmy Noether in 1918 (see Gold-

stein [27], Jos6 and Saletan [28], and Abraham and Marsden [29], and the references

therein). This issue is discussed in Chapter 2 in greater detail.

It is worthwhile mentioning that the Maxwell-Schrodinger equations are obtain-

able as the nonrelativistic limit of the Maxwell-Dirac equations


i>D = pmc2 D + ca [P qA/c]JD + q'TD (1.12)

A &
V2A A VV .A+ -4 (1.13a)
Sc2 c c

V2+ V A -47p (1.13b)
c









which are the equations of quantum electrodynamics (QED) [19, 24, 30]. Here the

wavefunction TD is a 4-component spinor where the first two components represent

the electron and the second two components represent the positron, each with spin-

1/2. The matrices 3 and a are related to the Pauli spin matrices [7, 8] and c is the

velocity of light. This system of equations possesses each of the symmetries of the

Maxwell-Schr6dinger equations and in addition is invariant under relativistic boosts.

1.3 Approaches to the Solution of the Maxwell-Schr6dinger Equations

Solving the Maxwell-Schr6dinger equations as a coupled and closed system em-

bodies the theory of radiation reaction [9, 26, 31], which is a main theme of this

dissertation. However, it should first be pointed out that (1.1) and (1.2) are com-

monly treated separately. In these cases, the effects of one system on the other are

handled in one of the following two ways:

* The arrangement of charge and current is specified and acts as a source for the
electromagnetic field according to (1.2).

The dynamics of the electromagnetic field is specified and modifies the dynamics
of the matter according to (1.1).

It is not surprising that either of these approaches is valid in many pli v-i, .1l situ-

ations. Most of the theory of electrodynamics, in which the external sources are

prescribed, fits into the first case, while all of classical and quantum mechanics in

the presence of specified external fields fits into the second.

As a further example of the first case, the dipole power radiated by oscillating

dipoles generated by charge transfer processes in the interaction region of p H

collisions can be computed in a straightforward manner [32, 33]. It is assumed

that the dynamics of the oscillating dipole is known and is used to compute the

dipole radiation, but this radiation does not influence the p H collision. As a

result energy, momentum, and angular momentum are not conserved between the

proton, hydrogen atom, and electromagnetic field system. As a further example of

the second case, the effects of stimulated absorption or emission of electromagnetic









radiation by a molecular target can be added to the molecular quantum mechanics as

a first order perturbative correction. The electrodynamics is specified and perturbs

the molecule but the molecule does not itself influence the electrodynamics. This

approach, which is known as Fermi's golden rule (see Chapter 3 and Merzbacher [7],

Craig and Thirunamachandran [34], and Schatz and Ratner [35]) is straightforward

and barring certain restrictions can be applied to many ]li ,-i, .1l systems.

The system of Maxwell-Schridinger equations or its relativistic analog can be

closed and is coupled when the Schr6dinger wavefunction T, which is the solution

of (1.1), is chosen to be the source for the scalar potential D and vector potential

A in (1.4). In particular, the sources of charge p and current J, which produce

the electromagnetic potentials according to (1.4), involve the solutions T of the

Schridinger equation according to


p = q4J* J =q{(*[-iV qA/c]i+ 1[iV qA/c]i*}/2m. (1.14)


On the other hand, the wavefunction T is influenced by the potentials that appear

in the Hamiltonian H in (1.8).

The interpretation of the Schridinger wavefunction as the source for the elec-

tromagnetic field was Schrbdinger's electromagnetic i. !; 11' .., -.:.. which dates back to

1926. The discovery of the quantum mechanical continuity equation and its sim-

ilarity to the classical continuity equation of electrodynamics only reinforced the

hypothesis. However, it implied the electron to be smeared out throughout the

atom and not located at a discrete point, which is in contradiction to the accepted

Born probabilistic or Copenhagen interpretation. Schridinger's wave mechanics

had some success, especially with the interaction of the electromagnetic field with

bound states, but failed to properly describe scattering states due to the probabilis-

tic nature of measurement of the wavefunction. In addition, certain properties of

electromagnetic radiation were found to be inconsistent with experiment.









Schridinger's electromagnetic hypothesis was extended by Fermi in 1927 and

later by Crisp and Jaynes in 1969 [36] to incorporate the unquantized electromag-

netic self-fields into the theory. That is, the classical electromagnetic fields pro-

duced by the atom were allowed to act back upon the atom. The solutions of this

extended semiclassical theory captured certain aspects of spontaneous emission as

well as frequency shifts like the Lamb shift. However, it was quickly noticed that

some deviations from QED existed [37]. For example, Fermi's and T., i. -'s theo-

ries predicted a time-dependent form for spontaneous decay that is not exponential.

There are many properties that are correctly predicted by this semiclassical theory

and are also in agreement with QED. In the cases where the semiclassical theory

disagrees with QED [37], it has always been experimentally verified that QED is cor-

rect. Nevertheless, the semiclassical theory does not suffer from the mathematical

and logical difficulties that are present in QED. To this end, the semiclassical theory,

when it is correct, provides a useful alternative to the quantum field theory. It is

generally simpler and its solutions provide a more detailed dynamical description of

the interaction of an atom with the electromagnetic field.

Since 1969 many others have followed along the semiclassical path of Crisp and

Jaynes. Nesbet [38] computed the gauge invariant energy production rate from a

many particle system. Cook [39] used a density operator approach to account for

spontaneous emission without leaving the atomic Hilbert space. Barut and Van

Huele [40] and Barut and Dowling [41, 42] formulated a self-field quantum electro-

dynamics for Schr6dinger, Pauli, Klein-Gordon, and Dirac matter theories. They

were able to eliminate all electromagnetic variables in favor of Green's function inte-

grals over the sources and were able to recover the correct exponential spontaneous

decay from an excited state. Some pertinent critiques of this work are expressed

by Bialynicki-Birula in [43] and by Crisp in [44]. Bosanac [45-47] and Dosli6 and









Bosanac [48] argued that the instantaneous effects of the self interaction are un-

ll,-i, .1] As a result, they formulated a theory of radiation reaction based on the

retarded effects of the self-fields. Milonni, Ackerhalt, and Galbraith [49] predicted

chaotic dynamics in a collection of two-level atoms interacting with a single mode of

the classical electromagnetic field. Crisp himself has contributed some of the finest

work in semiclassical theory. He computed the radiation reaction associated with a

rotating charge distribution [50], the atomic radiative level shifts resulting from the

solution of the semiclassical nonlinear integro-differential equations [51], the interac-

tion of an atomic system with a single mode of the quantized electromagnetic field

[52, 53], and the extension of the semiclassical theory to include relativistic effects

[54].

Besides semiclassical theory, a vast amount of research has been conducted

in the quantum theory of electrodynamics and matter. QED [19, 24, 30, 55] (see

Chapter 3), which is the fully relativistic and quantum mechanical theory of elec-

trons and photons, has been found to agree with all associated experiments. The

coupled equations of QED can be solved nonperturbatively [56, 57], but are most

often solved by resorting to perturbative methods. As was previously mentioned,

there are some drawbacks to these methods that are not present in the semiclassi-

cal theory. In addition to pure QED in terms of electrons and photons, there has

also been an increasing interest in molecular quantum electrodynamics [34]. Power

and Thirunamachandran [58, 59], Salam and Thirunamachandran [60], and Salam

[61] have used perturbative methods within the minimal-coupling and multipolar

formalisms to study the quantized electromagnetic field surrounding a molecule. In

particular, they have clarified the relationship between the two formalisms and in

addition have calculated the Poynting vector and spontaneous emission rates for

magnetic dipole and electric quadrupole transitions in optically active molecules.









In both the semiclassical and quantum mechanical context the self-energy of the

electron has been studied [62-65]. The self-energy arises naturally in the minimal

coupling scheme as the q4 term in the Hamiltonian (1.8). More specifically, the

electron's self-energy in the nonrelativistic theory is defined as


U f= d3xqp(x, t)'*(x, t)i(x, t) = dxfVd p(x', p(x'' ) (1.15)
x x'

As a result of the q4 term, the Schrodinger equation (1.10) is nonlinear in T. It

resembles the nonlinear Schrodinger equation [66]


ii -a(d2u/dx2) + b u2u (1.16)


which arises in the modeling of Bose-Einstein condensates with the Gross-Pitaevskii

equation and in the modeling of superconductivity with the Ginzburg-Landau equa-

tion.

In the relativistic theory, the electron is forced to have no structure due to

relativistic invariance. As a result, the corresponding self-energy is infinite. On the

other hand, the electron may have structure in the nonrelativistic theory. Conse-

quently, the self-energy is finite. The self-energy of the electron will be discussed in

Chapter 4 in more detail.

1.4 Canonical Formulation of the Maxwell-Schr6dinger Equations

The work presented in this dissertation [67] continues the semiclassical story

originally formulated by Fermi, Crisp, and Jaynes. Unlike other semiclassical and

quantum mechanical theories of electrodynamics and matter where the gauge is fixed

at the beginning, it will be emphasized that the gauge is a fundamental degree of

freedom in the theory and should not be eliminated. As a result, the equations

of motion are naturally well-balanced and form a well-defined initial value problem

when the gauge freedom is retained. This philosophy was pursued early on by Dirac,

Fock, and Podolsky [68] (see Schwinger [19]) in the context of the Hamiltonian









formulation of QED. However, their approach was quickly forgotten in favor of

the more practical Lagrangian based perturbation theory that now dominates the

QED community. More recently, Kobe [69] studied the Hamiltonian approach in

semiclassical theory. Unfortunately, he did not recognize the dynamical equation

associated with the gauge and refers to it as a meaningless equation.

It is believed that the Hamiltonian formulation of dynamics offers a natural and

powerful theoretical approach to the interaction of electrodynamics and wave me-

chanics that has not yet been fully explored. To this end, the Hamiltonian or canoni-

cal formulation of the Maxwell-Schrodinger dynamics is constructed in this disserta-

tion. (Canonicalmeans according to the canons, i.e. standard or conventional.) The

associated work involves nonperturbative analytic and numerical methods for the so-

lution of the coupled and closed nonlinear system of Maxwell-Schrodinger equations.

The flexibility inherent in these methods captures the nonlinear and nonadiabatic

effects of the coupled system and has the potential to describe situations where the

atomic and electromagnetic dynamics occur on the same timescale.

The canonical formulation is set up by applying the time-dependent variational

principle to the Schrodinger Lagrangian


sch iPV* [iV qA/c]J* [-iV qA/c]' Vj* q *J (1.17)
2m

and Maxwell Lagrangian together with a Lorenz gauge fixing term, i.e.,

x [/c+ V A]2
Max -'Max --
87 (1.18)
[-A/c- V(I]2 [V x A]2 [',/c + V. A]2
87 87
This yields a set of coupled nonlinear first order differential equations of the form


wri = OH/r1n


(1.19)









where w is a symplectic form, r7 is a column vector of the dynamical variables, and

H is the Maxwell-Schr6dinger Hamiltonian (see Chapter 4). These matrix equations

form a well-defined initial value problem. That is, the solution to these equations

is uniquely determined and known in principle once the initial values for each of

the dynamical variables ri are specified. These equations are also closed since the

Schr6dinger wavefunction acts as the source, which is nonlinear (see J in 1.14), for

the electromagnetic potentials and these potentials act back upon the wavefunction.

By representing each of the dynamical variables in a basis of gaussian functions GL,

i.e., r](x, t) = >c Gc(x)r q(t), where the time-dependent superposition coefficients

r]l(t) carry the dynamics, the time-dependent variational principle generates a hier-

archy of approximations to the coupled Maxwell-Schrodinger equations. In the limit

of a complete basis these equations recover the exact Maxwell-Schr6dinger theory,

while in any finite basis they form a basis representation that can systematically be

made more accurate with a more robust basis.

The associated basis equations have been implemented in a FORTRAN 90 com-

puter program [70] that is flexible enough to handle arbitrarily many gaussian basis

functions, each with adjustable widths and centers. In addition, a novel numerical

convergence accelerator has been developed based on removing the large Coulombic

fields surrounding a charge (that can be computed analytically from Gauss's law,

i.e., V E = -V24 = 47p, once the initial conditions are provided) by applying a

certain canonical transformation to the dynamical equations. The canonical trans-

formation separates the dynamical radiation from the Coulombic portion of the field.

This in turn allows the basis to describe only the dynamics of the radiation fields

and not the large Coulombic effects. The canonical transformed equations, which

are of the form Coj = aH/a0F, have been added to the existing computer program

and the convergence of the solution of the Maxwell-Schridinger equations is studied.









The canonical approach to dynamics enjoys a deep mathematical foundation

and permits a general application of the theory to many plr, -i, .,l problems. In par-

ticular, the dynamics of the hydrogen atom interacting with its electromagnetic field

has been investigated for both stationary and superpositions of stationary states.

Stationary state solutions of the combined hydrogen atom and electromagnetic field

-i,-1 I i as well as nonstationary states that produce electromagnetic radiation have

been constructed. This radiation carries away energy, momentum, and angular mo-

mentum from the hydrogen atom such that the total energy, momentum, angular

momentum, and charge of the combined system are conserved. A series of plots are

presented to highlight this atom-field dynamics.

1.5 Format of Dissertation

A tour of the Lagrangian and Hamiltonian dynamics is presented in Chapter 2.

Hamilton's principle is applied to the derivation of the Euler-Lagrange equations of

motion. Emphasis is placed on the Hamiltonian formulation of dynamics, which is

presented from the modern point of view which makes connection with symplectic

geometry. To this end, both configuration space and phase space geometries are

discussed.

In Chapter 3, the Schridinger and Maxwell dynamics will be presented from the

point of view of perturbation theory. In the Schridinger theory, the electromagnetic

field is treated as a perturbation on the stationary states of an atomic or molecular

system. In the long time limit, the Fermi golden rule accounts for stimulated transi-

tions between these states. As an example, the absorption cross section is calculated

for an atom in the presence of an external field. QED is discussed to emphasize the

success of perturbation theory. In the Maxwell theory, the electromagnetic fields

arising from specified sources of charge and current are presented. The first order

(electric dipole) multipolar contributions to the electromagnetic field are calculated.

Lastly, the bremsstrahlung from a gaussian charge distribution is analyzed.









Chapter 4 contains the main body of the dissertation, which is on the Hamilto-

nian or canonical approach to the Maxwell-Schrodinger dynamics. Nonperturbative

analytic methods are constructed for the solution of the associated coupled and

nonlinear equations. The gauge symmetry is discussed in detail and exploited to

cast the Maxwell-Schrodinger equations into a well-defined initial value problem.

The theory of canonical or symplectic transformations is used to construct a special

transformation to remove the Coulombic contribution to the dynamical variables.

The well-defined Maxwell-Schrodinger theory from Chapter 4 is numerically

implemented in Chapter 5. The associated equations of motion are expanded into

a basis of gaussian functions, which renders the partial differential equations as

ordinary differential equations. These equations are coded in FORTRAN 90. In ad-

dition, the (canonical transformed) equations associated with the Coulomb reference

are incorporated into the existing code. The dynamics of the spinless hydrogen atom

interacting with the electromagnetic field are presented in a series of plots.

A summary and conclusion of the dissertation are presented in Chapter 6.

1.6 Notation and Units

A brief statement should be made about notation. All work will be done in

the (1+3)-dimensional background of special relativity with diagonal metric tensor

ga = g"P with elements goo = goo = 1 and g11 = g22 933 = -1. All 3-vectors
will be written in bold faced Roman while all 4-vectors will be written in italics. As

usual, Greek indices run over 0, 1, 2, 3 or ct, x, y, z and Roman indices run over 1,

2, 3 or x, y, z. The summation convention is employed over repeated indices. For

example, the 4-potential A = (Ao, Ak) = (b, A) and A, = g,Av = (b, -A). The

D'Alembertian operator E = V2 92/9(ct)2 -= 2 is used at times in favor of 02.

Fourier transforms will be denoted with tildes, e.g., F is the Fourier transform of F.

The representation independent Dirac notation Ih) will be used in the discussion of

time-dependent perturbation theory, but for the most part functions h(x) = (xlh)







16

or h(k) = (kih) will be used. (It will be assumed that all of the functions of pil, -i,

are in C" and in L2 n L1 over either the real or complex field.) Since it is the

radiation effects present on the atomic scale that are of interest, it is beneficial to

work in natural (gaussian atomic) units where h = -1e| = m, = 1. In these units

the speed of light c + 137 atomic units of velocity.















CHAPTER 2
THE DYNAMICS

A dynamical system may be well-defined once its Lagrangian and associated

dynamical variables as well as their initial values are specified. This information

together with the calculus of variations [71] generates the equations which govern

the dynamics. Chapter 2 will detail the aspects associated with generating equations

of motion for dynamical systems.

Many different variational methods exist by which to generate dynamical equa-

tions, each having subtle differences [72]. However, all methods rely on the ma-

chinery inherent in variational calculus. Given a starting and ending point for the

dynamics, the calculus of variations determines the path connecting them. The dy-

namics is determined by extremizing (either minimizing or maximizing) a certain

function of these initial and final points.

In this chapter, the Lagrangian and Hamiltonian formalisms [27-29] are pre-

sented for discrete and continuous systems. The Lagrangian approach leads to sec-

ond order equations of motion in time, while the Hamiltonian or canonical approach

leads to first order equations of motion in time. The resulting dynamics are equiva-

lent in either case. However, the Hamiltonian approach enjoys a rich mathematical

foundation connecting differential geometry and dynamics [28, 29]. Much of the re-

mainder of this dissertation will be devoted to the canonical formulation of Maxwell

and Schr6dinger theories.

The time-dependent variational principle [73], which has its origin in nuclear

l.l, -i, [74], is the variational approach to the determination of the Schrodinger

equation. The Hamiltonian dynamics associated with the Schrodinger equation

evolves in a generalized phase space endowed with a Poisson bracket. With the









time-dependent variational principle, n.Iiii, -body dynamics may be consistently de-

scribed in terms of a few efficiently chosen dynamical variables (see Deumens et.

al. [75]). Additionally, the variational technology provides a means by which to

construct approximations to the resulting equations of motion in a systematic and

well-balanced way. As will be seen in Chapters 4 and 5, these approximations will be

of utmost importance in the numerical solutions of the coupled nonlinear Maxwell-

Schridinger equations.

2.1 Lagrangian Formalism

Before delving into a detailed account of Lagrangian dynamics it is instructive

to say a few words about the Lagrangian itself. The Lagrangian is a scalar function

of the vectors qk and $k (k = 1,..., N) with dimensions of energy. However, it is

not the energy nor is it ]li ,-i .11ly observable. The Lagrangian is a fundamental

ingredient in the determination of a dynamical system. That is, the dynamics of a

system may be known in principle once the system's Lagrangian is known and the

dynamical variables are given at some time.

The Lagrangian may have a number of symmetries. In 1918, Emmy Noether

(see Goldstein [27] and the references therein) proved that to each continuous sym-

metry there is an associated conservation law. For example, since all observations

indicate that Nature is invariant under time and space translations as well as spatial

rotations, so should be the Lagrangian. If the Lagrangian possesses time transla-

tion invariance, then the energy of the system is conserved. If the Lagrangian is

invariant to space translations (rotations), then the linear (angular) momentum of

the system is conserved. One last symmetry of significance in this dissertation is the

gauge symmetry. Since Nature is invariant to the choice of gauge, the Lagrangian

should maintain this symmetry as well. If the gauge symmetry is preserved, then

the system enjoys conservation of charge. Depending on the particular system at









hand, other symmetries may be of importance and should also be respected by the

Lagrangian.

2.1.1 Hamilton's Principle

Given a Lagrangian L(qk, k, t) dependent upon the N position vectors qk, the

N velocity vectors ck, and also the time t, the action I is defined by the path integral


I(qk, qk, t) = ftL(q k qk, t)dt k =1,..., N. (2.1)

That the variation of this integral between the fixed times tl and t2 leads to a

stationary point is a statement of Hamilton's Principle [27, 28]. Moreover, this

stationary point is the correct path for the motion. In mathematical symbols, the

motion is a solution of

sI = 6fLdt 0, (2.2)

where I1 is the variation of the action I. Only those paths are varied for which

6qk (t) = 0 = qk(t2). A particular form of the variational path parametrized by the
infinitesimal parameter a is given by


qk(t, a) qk(t, 0) + ark(t), (2.3)

where qk(t) q(t, 0) is the correct path of the motion and the vectors k (t) are

well-behaved and vanish at the boundaries ti and t2. By continuously deforming

qk(t, a) until it is extremized, the correct path can be found.
This parametrization of the path in turn parametrizes the action itself. Equa-

tion (2.2) may now be rewritten more precisely as

I(a)) = ) da 0 (2.4)
Oa a=o









which represents infinitesimal variations from the correct path. The calculus of

variations yields

aI(a) f2 L C,,' + L 9' i
Oa Jtl a 1 Oa (2.5)
OL C,1' t t2 (OL d OL O ,'
Oi' 9a t tJ A I (t m l ,a a '

where a partial integration was performed in the second line. Since 6qk(tl) = 0 =

6qk(t2), the surface term vanishes. The stationary point of the variation is therefore
determined by
t (9L d 9L 9'
f tdt}a -a -0. (2.6)
Atl a. dt ,' Al I =o I o
But since the vectors ',~ /0aC are arbitrary (choose in particular 0,' /0ac > 0 and

continuous on [t, t2]), the integral is zero only when

OL d OL
=0 (2.7)
c0,' dtc -,'

by the fundamental lemma of the calculus of variations. Equation (2.7) defines

the system of N second order Euler-Lagrange differential equations in terms of the

local coordinates (qk, k). Since these equations are valid on every coordinate chart,

the Euler-Lagrange equations are coordinate independent. It is demonstrated in

[28] that (2.7) can be written in a coordinate free or purely geometric form. If

these equations admit a solution, then the action has a stationary value. It is this

stationary value which determines the motion. The second order form of the Euler-
Lagrange equations can be seen be expanding the total time derivative to give

OL { 2 L 02 L 2L 0. (2.8)
-, 1i < + \ = 0. (2.8)

It will always be assumed unless otherwise noted that the Hessian condition is sat-

isfied. That is det{a2L/aq')' } / 0.

Lastly, notice that the Lagrangian is arbitrary up to the addition of a total time

derivative. That is, if L --L' = L + (d/dt)K for K a well-behaved function of the









dynamical variables, then the action

If t 6 L{L+ (d/dt)K}dt = K(t2) -JK(tl) + jtSi dt = SLdt = I6 (2.9)


since 6K(t2) = 0 = 6K(t). Thus the same Euler-Lagrange equations (2.7) are

generated for L' as for L. In other words, there are many Lagrangians that lead

to the same equations of motion. There is no unique Lagrangian for a particular

dynamical system. All Lagrangians differing by only a time derivative will lead to

the same dynamics. More generally, in the dynamics of continuous systems two

equivalent Lagrangians may differ by a purely surface term in time and space.

2.1.2 Example: The Harmonic Oscillator in (qk, k)

It is now useful to present a brief illustrative example. In two freedoms, the dy-

namics of a scalar mass subjected to the force of a harmonic potential with frequency

2Jk is determined by the Lagrangian (no summation)

1 1
L(qk k) 1 '"1k 1 2k/c
L(q -mLkq q k = 1, 2 (2.10)
2 2

which is a function of the real-valued vectors qk and qk. Application of the calculus

of variations to the associated action functional leads to (2.7) with OL/O,'' = /n'

and OL/o,' = -mcjq The second order Euler-Lagrange equations of motion are

-ni', +-m2 kq 0 k 1,2 (2.11)
2 2

with initial value solution qk(t) qk(to) cos(wckt)+ k (to) sin(wokt)/wc. It is said that

qk is an integral curve of the dynamical equation (2.11). Once the initial values

qk(to) and qk(to) are provided, the dynamics of the harmonic oscillator is known.

This dynamics occurs in a space whose coordinates are not just the qk, but both the

qk and qk. Some geometric aspects of this space will now be presented.



























Figure 2-1: The configuration manifold Q = 2 is depicted together with the tangent
plane TqkQ at the point qk E Q.

2.1.3 Geometry of TQ

In the Lagrangian formalism, the dynamics unfolds in a velocity phase space

whose points are of the form (qk qk). The position vectors qk lie in a differential

manifold called the configuration manifold Q, while the velocity vectors lie in the

manifold of vectors tangent to Q. The space formed by attaching the space spanned

by all vectors tangent to the point qk E Q is called the tangent fiber above qk or

the tangent plane at qk and is denoted by TqkQ. The union of the configuration

manifold Q and the collection of all fibers TqkQ for each point qk C Q (together

with local charts on TqkQ) is called the ,' /. ., :lu phase space, tangent bundle, or

tangent r-ui,;'-././ of Q and is denoted by TQ. It is that manifold that carries the

Lagrangian dynamics, not the configuration manifold Q. A picture is presented in

Figure 2-1 corresponding to the case where Q is the two-dimensional surface S2 of

the unit ball in IRa. The tangent plane at the point qk reaches out of S2 and into IR.

This larger manifold is where the associated Lagrangian dynamics occurs.









The integral curves of a dynamical system are vector fields and are called the

.;ii.ii,; or the dynamical vector fields. The velocity phase space dynamics is a

vector field on TQ denoted by AL -- (/Ol') + qk(a/ol,'), where qk and qk are

the components of AL and 0/c,1' and 0/0c1' form a local basis for AL. The time

dependence of a dynamical variable F(qk, k), which is an implicitly time-dependent

function on TQ, is determined by its variation along the dynamics. That is

kOF OF
F(qk A) L(F) (2.12)

The accelerations qk can be substituted directly from the dynamical equations.

Thus, the time dependence of a dynamical variable is determined by the equations

of motion themselves without even the knowledge of their solution.

Beyond functions and vector fields on TQ, there is another important geo-

metrical qii.,nili', called the one-form that is worth defining. One-forms on TQ

are linear functionals that map vector fields to functions. That is, if the one-form

S= A'.i, + AiJ, is applied to the vector field X = Xb(/qb) + X (a/a b), then

their inner product results in

(a|X) = A'X .l (/O qb) + AaX2dqa (a/qb) + A Xf.i l (O/ qb) + A2Xbdq4a(O/Oqb)

= AXb a + A2 b,6a
a I b a bI 2

= AXab + AXa

(2.13)

where dqa (O/qb) d a(O/O b) 6 and dqa(O/aqb) dqa(O/Oqb) = 0, and where

Aa and Xb are the local components, which are functions, of the one-form a and

vector field X. It is common to write (a|X) a(X). It should be pointed out that

the differential of a function is a one-form. That is

dF 8 F i F
dF =---i' '/ + F--'11- (2.14)
Ai~l I C~i









is a one-form and may be applied to the dynamical vector field AL to give

OF k OF k
dF(AL) (dF|AL) = .k F. (2.15)

The one-forms are also called covariant vectors or covectors and are dual to the

vector fields which are sometimes called contravariant vectors.

2.2 Hamiltonian Formalism

The Lagrangian formalism set up N second order dynamical equations which

required 2N initial values to fix the dynamics. Alternatively, and equivalently, the

dynamics may be described in terms of 2N first order equations of motion with 2N

initial values. This so called Hamiltonian dynamics evolves in a different tangent

manifold or phase space with generalized coordinates qg and Pa, which are governed

by the dynamical equations

OH OH
g and -a (2.16)
apa Oqa'

where the function H is called the Hamiltonian (see (2.18) below). It is itself a

dynamical variable and for many ]ll--i. .,1 systems it is the energy. Since (2.16) are

of first order, the associated trajectories are separated on the new phase space. The

change of variables from (qa, ga) to (qa, Pa) is accomplished by a Legendre transfor-

mation [27, 28]. The momentum conjugate to the vector qg is defined in terms of

the Lagrangian L by
Pa -a) (2.17)

Notice that this ,i: ,i,.(.i,/.: momentum is not a vector as is the velocity qa and does

not lie in the tangent manifold TQ. Rather the momentum pa is dual to the position

vector qg. It is a one-form and lies in the cotangent manifold T*Q. This difference will

soon be elaborated on. With the momentum pa and the Lagrangian, the Hamiltonian









function is constructed according to


H(qa,pa) = paq(q, pa)- L(qa,pa). (2.18)

Here it is assumed that the relation (2.17) can be inverted to solve for the velocity

g. Hamilton's canonical equations of motion (2.16), which are first order differential

equations in time, can now be obtained from an argument similar to that presented

in Section 2.1.1 on Hamilton's Principle. That is, if the Lagrangian in the action

integral (2.1) is replaced by L = pa0' H from (2.18), i.e.,


I(q Pa) ft[a (q, Pa)- H(qal)]dt, (2.19)

then Hamilton's equations follow in a straightforward manner.

2.2.1 Example: The Harmonic Oscillator in (qa,pa)

It is now useful to compare the Lagrangian and Hamiltonian dynamics for a

simple dynamical system. Recall the Lagrangian for the two freedom harmonic

oscillator in (2.10). That is

1 a 1 aa
L(q, ) 2-" q. --m gq q a = 1, 2. (2.10)
2 2

The momentum conjugate to qa is pa OL/Oq" = 11 and with it the Hamiltonian

becomes

paPq 1 q a Papa mrnwaqa
H(qaP) P 2rn 2rnwqq p-2r + (2.20)
2m 2 2 m 2

With this Hamiltonian the equations of motion are:

OH
a H _,
Opa (2.21)
OH
-Pa mc L

and have the initial value solutions qa(t) = q(to) cos(cwat) + ((to) sin(oat)/lla and

Pa(t) = Pa(to) cos(tat) mL~~ a(to) sin(cat). These are the integral curves of the first









order differential equations (2.21) and may be compared to those in the Lagrangian

formulation.

2.2.2 Symplectic Structure and Poisson Brackets

One of the many powerful aspects of the Hamiltonian or canonical approach

to dynamics is the flexibility and ability to treat positions and moment similarly.

This similarity among the coordinates is made explicit by the following notation:

"a qa a = 1,..., N
(2.22)
S= Pa-N a N+ 1,...,2N.

Similarly, the forces become OH/Op, OH/d"a+N and OH/Oq" OH/d"a so that

the equations of motion are:

= OH a= ,...,N
H (2.23)
-a a N+ 1,...,2N.

These Hamilton equations may be written more compactly as


ab =b (2.24)
aal'

where WJab are the matrix elements of the .i,,',/.', 1.+, form w. The symplectic form is

an antisymmetric 2N x 2N-dimensional matrix of the form


(ON N (2.25)
IN ON)

where ON and IN are the N x N-dimensional zero and identity matrices respectively.

The matrix (2.25) is also referred to as the canonical 1,,tiI,/ /., ,: form because it

satisfies the properties:

2 = -1 and wT -w, (2.26)


or equivalently wab bc = ac and LWab = -W-ba. The matrix element "ab with both

indices up is the inverse of Wab.









In (2.12), the time derivative or variation of a (implicitly time-dependent) dy-

namical variable F on TQ was demonstrated. In a similar fashion, F can be viewed

in the momentum phase space T*Q, which will be discussed shortly. It is

8. F F 8F 8 H
F(q, OF) = aH (2.27)
O ab Obw aa'

where the equation of motion (2.24) was inverted and substituted for Cb. The right

hand side of this equation is called the Poisson bracket of F with H. In general, it

may be written for any two functions in T*Q as

OF baG F G F G (2.28)
{F, G} a aca a (2.28)
O1b Oa O 9qa OPa OPa Oqa'

In particular, an alternative form of Hamilton's equations is derived when the Pois-

son bracket is applied to the coordinate That is


( {, H}. (2.29)

Since the Poisson bracket is bilinear, antisymmetric, and satisfies the Jacobi identity

{f, gh} = g{f, h + {f, g}h, the set of functions on T*Q forms a Lie algebra under
Poisson bracket {., .}. In fact, the Hamiltonian dynamics can naturally be studied

from this point of view [29, 73].

2.2.3 Geometry of T*Q

As was previously mentioned, the dynamics associated with Hamilton's equa-

tions of motion (2.24) do not unfold in the same velocity phase space TQ that was

defined in Section 2.1.3. These equations of motion define a vector field ( on a dif-

ferent phase space whose components are the functions wba"(H/la"). The integral

curves of this vector field are the dynamics.

Recall that the points of TQ are made up of qk and qk. The velocities qk are the

local components of the vector field q (0a/ I). However, the moment are the local

components of the one-form p.,,q"' = (OL/Oqa)dqa, which are not the components









of a vector field. Since one-forms are dual to vector fields, p.,dq"' lies in the dual

space of Tqa Q. This space is the cotangent space at qa and is denoted by T*~ Q. In

analogy with TQ, the cotangent bundle or cotangent r-i,;K'f;-./- T*Q is made up of

Q together with its cotangent spaces T* Q. Consequently, the carrier manifold for

the Hamiltonian dynamics is not TQ, but rather it is the phase space T*Q. The

dynamical vector field on T*Q is given by

_'bB 8 .a 8 9 H 9 9H 9
AH a a a a 0 a a aa (2.30)
Ob aqa Pa apa Opa Oqa Oqa pa'

where Hamilton's equations of motion (2.16) were substituted for the ja and pa.

There is one last geometric (Ii..,l il' that needs to be defined. The symplectic

form c is a two-form on T*Q. Two-forms are bilinear, antisymmetric forms that map

pairs of vector fields to functions. That is, if X = Xa(O/Oa) and Y = Yb(o/b)

are vector fields on T*Q, then

w(X, Y) = Xaybw(aO/a, /b) = Xa Yb = XaY YaXa. (2.31)

The matrix elements WLab = --ba are identical to those presented earlier. Since Lc is

nonsingular and the differential dc = 0, i.e., Lc is closed, the two-form Lc is called

a -.I,,i,'1 Il.:,1 form. In general, phase space is naturally endowed with a symplectic

form or structure. For this reason T*Q is also a -,/l, I ri~-, .: i;'f-, l,-l [29]. Lastly, it

should be mentioned that wc(X, Y) is a measure of the area between the vectors X

and Y. In fact, there is a powerful theorem attributed to Liouville [27-29] that states

that the phase space volume must be invariant under canonical transformations in

phase space. Canonical transformations are those transformations that maintain the

symplectic structure of Hamilton's dynamical equations

Sab OH
-abC -- (2.24)
a"a


More will be said on canonical transformations in Chapter 4.















CHAPTER 3
ELECTRODYNAMICS AND QUANTUM MECHANICS

The coupling of electrodynamics to charged matter is a complicated problem.

This complexity is compounded by the fact that the fields produced by charges in

motion react back upon the charges, thus causing a modification of their trajectory.

As mentioned in the introduction, the corresponding 1.li',-i. is often analyzed in

one of two ways. Either:

* The electromagnetic field is taken as an influence on the dynamics of the charges.

* The sources of charge and current are used to calculate the dynamics of the
electromagnetic field.

Chapter 3 will discuss both of these cases in detail. The first portion of this chapter

will set up the time-dependent perturbation theory which will be used to make

calculations in quantum mechanics under the influence of an electromagnetic field.

The second portion of this chapter will explore the electrodynamics resulting from

a given p and J. In particular, the multiple expansion will be introduced and used

to calculate the power radiated from an oscillating electric dipole. Additionally, the

electromagnetic fields corresponding to a gaussian wavepacket will be presented. In

the narrow width limit of the gaussian, the resulting pilr, -i, reduces to the expected

textbook results for a point source.

3.1 Quantum Mechanics in the Presence of an Electromagnetic Field

The dynamics of charges in an external electromagnetic field may be studied

at varying levels of sophistication from a purely classical description of both charge

and field to a fully quantum treatment. Various semiclassical or mixed quantum-

matter/classical-field descriptions are available as well as fully quantum and rela-

tivistically invariant treatments such as quantum electrodynamics.









Time-dependent perturbation theory [7] is a systematic method by which to

calculate (among other things) properties of the dynamics of charges in an external

electromagnetic field. In this section, the time-dependent perturbation theory is in-

troduced for a general perturbation in the context of quantum mechanics. Emphasis

is then placed on the classical electromagnetic field as a particular time-dependent

perturbation V. Within this framework the perturbation is seen as causing tran-

sitions between two stationary states |Bk) and ITm) of an atomic system, and is

symbolized to lowest order by the matrix element Vkm = ('k VITm). Experimen-

tal observables such as the rate of transition or absorption cross section may be

calculated from Vkm. Additionally, time-dependent perturbation theory gives a pre-

scription for calculating successively higher order corrections to Vkm, which may

in turn provide better and better agreement with experiment. This section con-

cludes with a discussion of quantum electrodynamics, in which both matter and

fields are quantized and the description is relativistically invariant. Here again the

time-dependent perturbation theory (often in the form of Feynman diagrams) is the

essential machinery used in calculations.

3.1.1 Time-Dependent Perturbation Theory

An important class of solutions to the Schrodinger equation (1.1) are those

which are eigenfunctions of the Hamiltonian operator H. These solutions Im) satisfy

the time-independent Schr6dinger equation


H|ITm) Em Im) (3.1)


and are called stationary states. A general solution |1(t)) of the Schr6dinger equa-

tion (1.1) may be constructed from these stationary states according to


I'(t)) e-iH(t-to)I(tO)) 0- m-iEm(t-to) 'lm)IT'm I(to)),


(3.2)









where I|(to)) is an initial state vector and where the sum over m may imply inte-

gration if the energy spectrum is continuous.

Equation (3.2) is only applicable when the Hamiltonian is time-independent.

For if H H(t), then the energy of the system is not conserved and H admits no

strictly stationary states. However, it may be possible to split a time-dependent

Hamiltonian into the sum of two terms:

H Ho + V(t), (3.3)

where Ho is time-independent and describes the unperturbed system while V(t)

accounts for the time-dependent perturbation. To fix ideas, consider for example

the electronic transition induced by a passing electromagnetic disturbance that is

localized in both space and time. In other words, the system is initially unperturbed

for some long time and is in an eigenstate of Ho. While in the interaction region

the system is perturbed by V after which it settles down into another unperturbed

eigenstate of Ho for a sufficiently long time.

Time-dependent perturbation theory seeks to connect the stationary states of

the unperturbed system, i.e., those states -.,i i- vii -:


Ho0 m) = Em\ m), (3.4)

with the time-dependent perturbation V(t). These calculations are most clearly

demonstrated in the interaction picture. In the interaction picture the perturbation

is singled out by applying the unitary operator Uo = exp(iHot) to I|(t)). That is


II(t)) = eiHotl(ti ) (3.5)

and the time-dependent Schr6dinger equation (1.1) becomes


(3.6)


4d1dt)11P1(t)) VIMITIM),








where Vi(t) = UoV(t)U In other words, the interaction picture separates the
plr, l-i. that depends upon the perturbation from the pir, -i. that depends upon
only the unperturbed system. The state vector at time t is obtained from that at
time to via

I|P (t)) = U,(t, to)l I(to)), (3.7)

where Ui is the time evolution operator which satisfies

Ui(t, to) 1 if dt' V(t')Ui(t',to). (3.8)

The time evolution operator connects the (orthonormal) stationary states k}) and

TIm) according to

('kUi(t, to)lIm) 6k- if dt'(kIV(t')Ui(t',t o

=6km iZJt dt'(kVi nn(t')I)( Ui(t', to) m (3.9)

= 6km i- n dtI iW-k' ('Ilk I V(t) I'n) I UI(t', to) I'm,),

where |,))(,1 = 1 and Ikn = Ek E, were used.
The time-dependent perturbation theory is now set up by iterating on (3.8). If
the perturbation V is small then the time evolution operator becomes a power series
in V. That is
I tl + ( t f t
U,(t, to) 1 dt' V(t') (-i)2f dt dt Vt"(t")+. (3.10)

And so at first order the transition amplitude between two distinct states of energy
Ek and Em (with k $ m) is


('k IUi(t, tO) m I'd iw Ik V(t') -f dtm)' '


(3.11)









Assuming that the perturbation is sufficiently small, the probability of finding the
-v-,1 IIi in the state Ik}) is given by
2 2 t 2
Pk-m (t) 'k I \Ui (t, to) 2 dto'' k V m (t) 2). (3.12)

If the perturbation is localized in time then to and t may be naively extended to
infinity to yield the transition probability

Pk_,,m(+o ) ('' U1(+o ,- )|) -) if dteI m(Vi(t) Im}) (3.13)

which involves a Fourier integral of the matrix element Vkm (= 'I'V(t)lm).
3.1.2 Fermi Golden Rule

The formalism set up thus far is also applicable for time-independent perturba-
tions V 1 V(t). In this case the transition probability can be obtained from (3.12)
as
,1 cos(km~ t)
Pkivm(t) = 21('v k ~ m) 2 1 o(- ) (3.14)
(Ek Em)2
which is proportional to t2 if Ek a Em.
Now consider the situation in which there is a near continuum of final states
available having energies in the interval (Em AE/2, Em + AE/2). If the density
of the near continuum states is denoted by pF(E), then the transition probability
to all of these states is given by
Z1.p+im/t f> -E cos(wivmt)
keFPkTnt) Em -E/2 2|mI| T|)|2 COsk) PF(Ek)dEk, (3.15)

where the sum runs over all states Ik}) belonging to the near continuum of final
states. The quotient [1-cos(wkmkt)]/(Ek-Em)2 is sharply peaked at Ek = Em which
confirms that the dominant transitions are those that conserve the unperturbed
energy. Since both I(k IV1Im) 2 and pp(Ek) are approximately constant around Em
and t is such that AE > 2r/t (i.e., long time behavior), the transition probability









becomes

ZkeFPk,-m(t) R 2 k( V'm) 2 PF(Ek_) 1 2 co kt) dk
tkm (3.16)
27rt IVI)P F 12 P(Ek)

which increases linearly with time.

The total transition probability per unit time or transition rate F is given by

F (d/dt)EkeFPkm(t) 27 %Ik VI'm) 2PF(Ek) (3.17)

and is constant. Fermi's golden rule of time-dependent perturbation theory [7, 34,

35] embodies the tendency for the perturbed system to make energy conserving tran-
sitions for which the probability increase as t2 or to make nearly energy conserving

transitions which oscillate in time. Either way the transition rate F is constant.

Fermi's golden rule may be extended to include perturbations that vary har-

monically in frequency Lc. An electromagnetic disturbance of a charge would be an

example. In this case the golden rule generalizes to

P= 2|(W'Vk mV F) 2PF(E+n +c). (3.18)

3.1.3 Absorption of Electromagnetic Radiation by an Atom

Recall the electromagnetic field coupling to quantum mechanics is given by the

minimal ((wplinlII prescription

i(9/9t) i(9/9t) q#
(3.19)
-iV -iV qA/c,

where A" = (4, A) are the dynamical variables of the electromagnetic field. Apply-

ing this transformation to the Schr6dinger equation iI = P2 1/2m + Vo results in

the Schr6dinger equation coupled to the electromagnetic field

[P q qA/c]2
i [ = P iqA + VoT + qKD (3.20)
2m









with Hamiltonian
[P qAlc]2
H + o + q p. (3.21)
2mr

In the Coulomb gauge (see Appendix A for details) this Hamiltonian becomes

P2 +V qA q 2 2
H = o A P + A2. (3.22)
2m mec 2mc2

The external free electromagnetic field evolves according to V2A A/c2 = 0 with

V-A = 0 and D = 0 since it is assumed that the charges do not themselves contribute

to the field. By ignoring the quadratic term in A, the Hamiltonian H separates into

an unperturbed portion
P2
Ho = + V (3.23)
2m

plus the perturbation

V = A P. (3.24)
mec

It should be pointed out here that substantial confusion has existed in the litera-

ture over the A P appearing in the perturbation V. This confusion was due the

widespread use of E r and its higher order approximations [59, 60] instead of A P.

The relationship between these two approaches have been thoroughly investigated

in [76-79].

The cross section for stimulated absorption (or emission) of radiation by an

atom may be calculated via Fermi's golden rule. If the external field varies harmon-

ically in frequency as a plane wave, then the perturbation becomes

V(x, t) q {Aoei(x-w) + Ae-i(kxw) }. P (3.25)
mc

where c is the field's polarization. The rate of energy absorption by the atom is


Lw = |Ao 2 (^,ik-x6 P m) 2pp(Em +). (3.26)
m2c2









If the density of the near continuum states is narrow then


PF(Em + c) = 6(Ek E + ) = 6(ckm + c) (3.27)

and the absorption cross section Uabs(bc) r/Jo becomes
2
q2 Aol2 (1, kx. P lIm,) (cck, +c)/m2c2
abs ) ccAo 2/2rc (3.28)

41 22 2 2
4' |Ao 2 ik-x6 P ,m) 6(cckL + ),

where 1o = A|Ao 2/27rc is the incident flux of photons of frequency Lc. Similarly the

emission cross section is


aem(T ) 1Ao 2 (' ,-ikxe P'm) (cckm ). (3.29)


Notice that the time-dependent perturbation theory gives properties of the so-

lution but not the solution. That is, the cross section is easily accessible but the

wavefunction and 4-potential are not. The cross section is a property of the solution

and can be calculated from knowledge of the solution. Of course, the wavefunction

and 4-potential constitute the actual solution.

3.1.4 Quantum Electrodynamics in Brief

The quantum theory of electrodynamics [19, 24, 30], also known as QED, is the

interacting quantum field theory of electron and photon fields. The relativistically

invariant QED is one of the most successful ]li', -i .,1 theories to date, in that there

is no evidence for any discrepancy between experiment and prediction. However,

QED is beset by many mathematical and logical difficulties. These difficulties are

in some cases avoided by 1.li,--i .1l arguments or simply concealed from view as in

the renormalization of mass and charge.

Putting aside its inconsistencies, QED is a prime example of the success of

time-dependent perturbation theory. A combination of the free Dirac theory and

the free Maxwell theory provide the unperturbed states on which the interaction









Lint = -JAJ /c operates. The free QED Lagrangian density

1
free QED ='[' "' mc]y IF F lF" (3.30)
16-

gives the equations of motion for the free electron


,' mc] = 0 (3.31)

and the equations of motion for the free electromagnetic field

1,F"" = 2A" O"( A) 0", (3.32)

where the Dirac 7-matrices are related to the Pauli spin matrices (4.66), = toy7

is the Dirac adjoint of the four component spinor y, and F, = 0,A, 0,A, is

the electromagnetic field tensor. This noninteracting theory sets up the free un-

perturbed in-state pi .. p,)in and out-state ki .. ki)out, which will be connected

by Ui(+oo, -oo) = T{exp[ifd4xinat]}, where T is the time-ordering operator. The

resulting matrix elements will yield some properties of the dynamics.

Working in the interaction picture, the machinery of time-dependent perturba-

tion theory is used to construct the scattering matrix or S-matrix

out(ki ... kmlPIl... p,)in "i la(ki)... a(k,)Ui(+oo, -oo)at(p) ... at(p,) )n

(3.33)

where S = 1 + iT and |Q)in is the in-vacuum. The respective fermion creation

and annihilation operators at(k) and a(k), create and annihilate single fermions of

momentum k according to at(k)l|) = Ik) and a(k)l|) = 0 where the spin has been

neglected for simplicity. The situation in which the particles do not interact at all

(the 1 part) as well as the interesting interactions (the iT part) are both included in

the S-matrix. The interacting components are commonly collected and are referred









to as the T-matrix. Together 1+ iT is used to define the invariant amplitude M as

out(ki ... k |UI(+oo, -oo)|pi p,)in

= (27)4'(pl + + p pn 11 k ). 0(km) -iM(P,-"", P,n ki,-" ,km).

(3.34)

This invariant amplitude is analogous to the scattered wavefunction of quantum

mechanics, i.e.,

ilk(X) N [ik-x fk()k l (3.35)

where the unscattered field exp(ik x) and the spherically scattered field exp(ikr)/r

are indicated schematically in Figure 3-1. In fact all of quantum mechanics is


I
T 'J-" Scattered field
I -
I \ / /
Incident field


Figure 3-1: The coefficient 1 of the unscattered plane wave exp(ik x) is analogous
to the 1 part of the S-matrix, while the scattering amplitude fk(Q) which modulates
the scattered spherical wave exp(ikr)/r is analogous to the iT part.


just the nonrelativistic limit of QED. Unfortunately, while QED is suitable for the

scattering of single particle states to single particle states, it requires great effort to

deal with bound states.

The probability of finding ki ... k)Out in |pi .. p)'in is given by

2 13ki d3 km
P(+oo)~ out{ki...km|pi...P.) ...
P( ) gUi(k kl l p)pin 2 d3d31.1 d313 d3k

= nl(a(ki)... a(k)Ui(+oo, -oo)at(pi)... at(p),)l 2 d31.. d3krv
(2) (3.36)
(3.36)









which is analogous to (3.13). A similar connection can be made in the cross section.

If n = 2 in the in-state, then the differential scattering cross section d( becomes

2 d3ki d3k
do- ~ iM(pi,p2 -,- ki,-- k) 2- ..- (3 (27r)46(pl +2 ki ... knm)

(3.37)
which is analogous to (3.28). As in quantum mechanics, time-dependent pertur-

bation theory in QED gives a prescription by which to calculate properties of the

solution which rely on scattering amplitudes, e.g., cross sections, decay rates, and

probabilities. It is considerably more difficult to compute the actual solution, which

in this case would be the states on which the field operators
3k e-ik-x ,ik-x
A,(x, t) = f (2-) (k, A)c(k, A) + c(k, A)ct(k, A) k}
d3k e-ik-x vikx (3.38)
.'. (x, t) = Jv d()3 k ui(k, s)(k, + v(k, s)bt(k, s) }

act. In (3.38), c, and are helicity eigenstates of A,, and {c, ct} are photon creation

and annihilation operators. Similarly ui and vi are eigenspinors of and {a, at}

and {b, bt} are electron and positron [80] creation and annihilation operators.

Lastly it should be pointed out that a beautiful representation of the time-

dependent perturbation theory was introduced by Feynman [55]. These so called

Feynman dinqrlm~n provide a pictorial version of the invariant amplitude

iM = C out(kl kmUi(+oo, -oo)lpl p,)in

SC out(ki.. k T{eT d4 ij nt P.. Pn (3.39)

S[out (k... km| (1 + d4 T{j } + ) pl pn conned
L J connected









For example, the invariant amplitude for Bhabha scattering, i.e., e+e- e+e-, is

iMBhabha [ut (k1k2l pip)ll2 + out(klk2 jd.T int4 L} p1p2)i + O( )]nnt

1'- k2 1- k2


+ + O(L1nt)


P1 k1 Pl ki (3.40)

where each of the above diagrams corresponds to a term (or portion thereof) in

the perturbative expansion of iMBhabha. These tree order diagrams are the lowest

order nonvanishing diagrams that contribute to and are the largest part of the

Bhabha scattering invariant amplitude. Higher order perturbative corrections to the

amplitude also have pictorial representations and may be systematically constructed

using Feynman's prescription.

In this manner the time-dependent perturbation theory may be diagrammati-

cally written to any order, translated into mathematical expressions, and computed.

While this is by no means an easy task, the invariant amplitude may in principle be

calculated to any order. Notice again that this machinery produces the amplitude

iM, which is a property of the solution but not the actual solution.

3.2 Classical Electrodynamics Specified by the Sources p and J

If the sources of charge and current are known, then the dynamics of the result-

ing electromagnetic field can be calculated from Maxwell's equations at each point

in space-time. These fields may behave quite differently depending on the motion

of their source. For example, a static source gives rise to a purely electrostatic field,

while a uniformly moving source creates both an electric field and a magnetic field.

More importantly, if the source is accelerated then electromagnetic radiation is pro-

duced. Electromagnetic radiation is a unique kind of electromagnetic field in that









it carries away energy, momentum, and angular momentum from its source. The
radiation field is not bound to the charge as are the static fields.
In this section the electromagnetic fields produced by an oscillating electric
dipole are calculated to lowest order via the multiple expansion. The correspond-
ing power and radiation pattern are also presented. Then, the dynamics of the
bremsstrahlung produced by a wavepacket source is analyzed. It is shown that the
wavepacket's fields reduce in the narrow width limit to the usual point source results.
The consistent coupling of electrodynamics and quantum mechanics is needed
because the sources of charge and current produce electromagnetic fields and these
fields act back upon the sources. The understanding of this process requires the
inclusion of recoil effects on the charges due to the electromagnetic field. These
effects, known as radiation reaction effects, are a main aspect of this dissertation
and will be discussed in detail in Chapter 4.
3.2.1 Electromagnetic Radiation from an Oscillating Source
In this section the Lorenz gauge (see Appendix A for details) is used to inves-
tigate the electromagnetic radiation produced by a localized system of charge and
current [9] which vary sinusoidally in time according to (the real part of):

p(x, t) = p(x)e-i
(3.41)
J(x,t) = J(x)e- i

It is assumed that the electromagnetic potentials and fields also have the same time-
dependence. The general solutions to the wave equations of (A.5) are given by


4(x, t) = fVd3x'fr dt' t6 (t [t + IX X
(3.42)
A(x, t) =-jIVd3x'f- dt'J(x, t') 6(t' [t+X -
c |x x'k c
where G(+)(x, t; x', t') = 6(t' [t + x x'l/c])/47rx x'| is the retarded Green's
function for the wave operator 02 = -D 2/9(ct)2 V2 (see Appendix B). It is









assumed that there are no boundary surfaces present. With the oscillating sources

from (3.41), it will be seen that all of the dynamics of the electromagnetic field for

which ac / 0 can be described in terms of the A alone. The component of the

electromagnetic field for which c = 0 is just the static electric monopole field



The vector potential for all other frequencies is

1 eiklx-x'l
A(x) Ifd3x' J(x'), (3.44)
c X x'|

where the wavevector k = c/c and it is understood that A(x, t) = A(x)e- it

For a given charge density J, (3.44) could in principle be computed. With the

resulting vector potential the electromagnetic field may be calculated from Amp6re's

law. That is


B V x A (3.45a)

E -V x B (3.45b)
k

in a region outside the source. Instead of evaluating (3.44) exactly, general properties

of its solution may be determined whenever the dimensions of J are much smaller

than a wavelength. That is, if the dimensions of the charge density are of order d

and the wavelength A = 27/k, then d < A. From these distances, the following three

spatial regions may be constructed:

The near or static zone: d < r < A

The intermediate zone: d < r ~ A

The far or radiation zone: d < A < r

In each region the electromagnetic field behaves quite differently. For example in

the near zone, the fields behave as if they were static fields which show strong









dependence on their source. On the other hand in the far zone, the fields display

properties of radiation fields which are transverse and fall off as r-1.

The static near zone fields may be obtained from (3.44) by noting that kr < 1

since r < A. In this case exp(iklx x'l) ~ 1 and the vector potential becomes

Near (X) d, J(x')
Anear (x) = 4 d3x '
C X x'

C'f- 1 l+1( () (') } (3.46)
ikp
r
where Ix x' -1 has been expanded into the spherical harmonics Ym and an inte-

gration by parts was performed with all surface terms vanishing. The equation of

continuity iap + V J = 0 was also used in the computation as well as the definition

of the dipole moment p = fvd3xp(x). From (3.45), the resulting magnetic and

electric fields are:

ik
Bnear n x p ... (3.47a)
3n(n f p)- p
Enear = +( p) (3.47b)

where f is the unit vector in the direction of the observation point x. Notice that

Enear is independent of the frequency ac and is thus purely static. As expected Bnear
is zero in the static limit Lc 0. A multiple expansion of the near zone vector

potential can now be made and successively better results may be obtained by going

to higher orders in (1, m).









At the other extreme, the far zone fields for which kr > 1 may be obtained

from (3.44) by noticing that

x- x'= V(x- x'). (x- x')

2x x' Ix'12
1x|2 IX12
xl 1 X -(3.48)
x( x X

T-- -^X
= r n -xX

since x' < |x| = rfil = r. A picture of the corresponding situation is shown in

Figure 3-2, where the x'-integration runs over the domain of the source J. With the
x


n 'r= |x|
d
O

Figure 3-2: In the radiation zone, the observation point x is located far from the
source J. In this case the distance Ix x| w r fi x'.

approximation (3.48), the far zone vector potential becomes

ikr 1
Afar(x) dz'J(x')e-7 fx'
1ikr (_ikm 1
f d3Xj (Xi)"(ikn jd ,J(x)()fi x)Fr (3.49)
SBf k2!
m=0

rikr
-ikp-- +.. ,

where Ix x'|-1 m r-1 if only the leading term in kr is kept. It can now be seen

that the vector potential is an outgoing spherical wave with mth-angular coefficient

fyd'J(x'x)(-ikni x')m/cm!. From (3.45), the corresponding fields are:

Bfar = k2-- (Ix p) 1- "+ (3.50a)

Eaxik [=1 i-X- ikre[3ii(i p)- p] + -- (3.50b)









The magnetic field is transverse to the radius vector x = ri while the electric

field has components longitudinal and transverse to x. Both fields fall off like r-1

at leading order. The r-l-fields are the true radiation fields which carry energy,

momentum, and angular momentum to infinity. This can be seen from the time-

averaged differential power radiated per unit solid angle

dP 1 2fi C
S Re rni -Ex B*
dQ 2 47r
-c i xP2 (3.51)
87w
ck4
Ip12 sin2 0
87
which in this case is a measure of the energy radiated per unit time per unit solid

angle by an oscillating electric dipole p. Integrating this expression over 2 = (0, 0)

gives the total power radiated, i.e.,

dP ck4
P = d Ip12 (3.52)
dQ 3

The corresponding radiation pattern is shown in Figure 3-3. In general, the power

radiated by an i-pole goes like k2(l+1). Notice that it is the r-l-fields whose power

makes it to infinity in three dimensions. This is because E x B ~ r-2 which exactly

cancels the r2 in the measure factor d3x = r2drd. In two dimensions, it is the

r-1/2-fields whose power makes it to infinity since d2x = rdrdO. As before, a more

accurate description of the radiation field is obtained by including higher order terms

in the sum (3.49). The lowest order (nonvanishing) multiple contributes the most

to the field.

In the intermediate zone, neither of the previous approximations are valid. In

fact all terms in the previous series expansions would have to be kept. The under-

standing of the behavior of the fields in this zone requires the more sophisticated

machinery of vector multiple fields. The interested reader is referred to [9] for a

detailed discussion of multiple fields of arbitrary order (1, m).










X



















Figure 3-3: The differential power dP/dQ or radiation pattern corresponding to
an oscillating electric dipole verifies that no radiation is emitted in the direction
of the dipole moment. Rather the dipole radiation is a maximum in the direction
transverse to p.

Outside of the pll-i literature there is also a large amount of engineering

literature in the field of computational electrodynamics. In this area, Maxwell's

field equations are often solved numerically by finite element methods (see Jiao and

Jin [81] and references therein). Many applications of this work lie in electromagnetic

scattering, waveguiding, and antenna design. The inverse source problem [82] is also

another area of interest in engineering. Here, the goal is to determine the sources

of charge and current with only the knowledge of the electromagnetic fields outside

of the source's region of support. This problem has benefited from the work of

Goedecke [83], Devaney and Wolf [84], Marengo and Ziolkowski [85], and Hoenders

and Ferwerda [86], who have demonstrated the decomposition of the electromagnetic

field into nonradiating and purely radiating components.









3.2.2 Electromagnetic Radiation from a Gaussian Wavepacket

Consider the gaussian wavepacket with initial position r moving with constant

velocity v

''(x, t) = 2 3/4 l e2[x-(r+ t)]2 iVX (3.53)

where b = 1//2 is the wavepacket width. The corresponding probability current

is given by

J(x, t) = vp(x, t) -= [*(-iVW) + W(iV *)]. (3.54)
2m
In Fourier space this current becomes J(k, t) qv exp(-ik [r + vt] k2/ -) and

the vector potential is obtained by integrating against the Green's function D() for

the wave operator (see Appendix B). The vector potential becomes

A(x, t) = f evd3kik-x fo dtD(+) (t, t')47cJ(k, t')
)3A(xt) -

S4cqv d3k eik(x-r)-k2 82 o ,(t t') sin ck(t t') -ikvt (3.55)
-47cqvf V (2) r) 0 ckt, eC- (355)
(27)3 ck
d3 k eik [x-(r+vt)]-k2 /82
47cqvfV
47 (2r)3 C2k2 (k. v)2

which is difficult to perform analytically due to the complicated angular dependence

of the integrand. For nonrelativistic velocities, A can be approximated by

d3k eik-[x-(r+vt)]-k2 /f2
A(x, t) 4cqv (2 c [ (v/c) cos
(2)3 C2k2 1 kUC)2 COS2 0
2 qv d ekik.[x-(r+vt)]-k2 /f2 (3.56)
2 7 C-- k--;
qv erf(V/2 x -(r + vt)|)
c |x (r + vt)

where Gradshteyn and Ryzhik [87] was used. The norms of this vector potential and

its associated current density J are plotted along the x-axis in Figure 3-4 for two

different velocities. The charge q is taken to be negative. Notice that A follows the

charge distribution and that A will generate an electromagnetic field. For v/c < 1

this result is equivalent to a Galilei boost of the fields from the rest frame of the







48

I I f I




0.5








-5 0 5 10 15 20
x

Figure 3-4: The norms of J and A are plotted with different velocities along the
x-axis.
source. Only the electrostatic field remains by going to the rest frame. And so,

there is little difference between uniform motion and no motion.

As stated previously, the more interesting field dynamics is created whenever

the source is accelerated. To this end, consider the vector potential arising from a

moving charge whose current has the simple time dependence

J(k, t) qv(t)e-ik*(r+vt)-k2 /82
(3.57)
ve-ik-(r+vt)-k2/8 ( t)(t1 t),

where v is constant. This time dependence corresponds to a situation in which a

source is suddenly accelerated from a standstill to a uniform movement with velocity

v and is then instantaneously decelerated again to a standstill (see Figure 3-5). In

each of the three temporal regions of the current, the vector potential has a different









ct

- - - .





x


I ,. I- II
--------------------





Figure 3-5: The trajectory or world line r(t) of the charge is plotted. Electromag-
netic radiation is produced at ti and t2 and moves out on the light cone.

behavior. Obviously for t < to, A(x, t) = 0. For to < t < ti,


A (x, t) -
A(x,t) qv qverf(/ x -(r vt)
c ix (r +vt)|
I erf( [c(t to) x- (r + vto)) (3.
2 x (r + Vto)
1 erf(v2[c(t to) + Ix (r + vto)1])
2 |x (r + vto) I

and for t > ti,

S qv erf(v [c(t to) x- (r + vto)])
A(x, t) -x
2c |x (r + Vto)
erf(2-[c(t to) + x (r + vto) ])
Ix (r + vto)l
(3.59)
erf( 2[c(t t1) + Ix- (r + vtl)|])
x (r + vti)
erf( 2[c(t tl)- x (r + vti)])
ix -(r + vti)|

Again nonrelativistic velocities are assumed. A space-time plot of the norm of this

piecewise vector potential is shown in Figure 3-6. Note that the charge was at rest

until the time to, where it was instantaneously accelerated to a velocity of magnitude












2

0

-2

-10

-5

x


10 0



Figure 3-6: The bremsstrahlung radiation from a charged gaussian wavepacket
moves out on the smeared light cone with maximum at x = ct.

v. Then the charge moved uniformly with v until the time ti, when it was instan-

taneously decelerated to rest again. Since electromagnetic radiation is produced

whenever the velocity changes in time, electromagnetic ripples are produced at to

and ti. The ripples move out as radiation at the velocity c of light. Figure 3-6 shows

the light cone, which is smeared out due to the nonpointlike structure of the charge.

The vector potential presented so far has both longitudinal and transverse com-

ponents. For the time being, the tranversality of the A is not important. It turns

out that the only fields which contribute to the Poynting vector or to the power are

the transverse fields. And so it does no harm to keep the full vector potential. For

the interested reader, the transverse vector potential AT associated with (3.58) and

(3.59) is calculated in Appendix C by analogy to the quadrupole moment tensor.








The electric and magnetic fields corresponding to (3.59) are


E ~A
c
qvf
c


2 e-2 [c(t-to)- R(to)]2
VxL R


_ -2[c(t-to)+R(to)]2
(to)


(3.60)


S_2-2[c(t-2 t)+R(t_)]l2 C-22[c(t-t1)- R(t)]2
+ R(t) I
qvc (\
v [g )h (t )
h- (t)


neglecting the purely longitudinal -V(I, and

B VxA


/ v x [I(ti)g (t)


where R(t)


= |R(t) I= x- (r+vt)| and where the unit vectors u(to)


R(to)/R(to)


and u(ti) = R(tl)/R(ti). With (3.60) and (3.61), the Poi,,l'I.,; vector is


S ExB
47r
-"2 h (t) v x {vx [u(t1)g+(t)


The differential power radiated into the solid angle d( at time t becomes

dP(x, t)
P(x, t) R(t)2 n S(t)
dQ


(v* (to))g0(t)] n. v


(3.63)


-t)+R(ti)]2 -22[c(t-tl)-R(t)]2
R(tl)


- ,-[ -to)-R(to)R ]2 --- I,
R(to)


-to)+R(to)]2
I


(3.61)


(3.62)


qv I [i(tl)e-2e2[c(t-
c VT


fi(to)90 ()],


Sf2- (t) (t) 2
27-(2ffl


V2 [( u(tl))9gl(t)- ( U(t))90+(t)]

27 c 9(( () (v x (x) (V x (to))]
+(t) [(v x 1i) (V x l-(tl))]}


u(to )] ) .


[(V -(tl))gl+(t)









where the unit vector i = R(t)/R(t) is normal to the surface of the ball that

emanates from the radiation source. The vectors v, f, i(to), and u(ti) are all

constant in time. By choosing the z-axis along the velocity v, the angles between

v and the unit vectors i(to) and u(ti) are 60 and 61 respectively. With a little

geometry, it can be verified that

sin 0 sin 0
sin 60o and sin 61 (3.64)
\1+ (v/ct)(ti to) cos0 \/1 (v/ct)(t to) cos 0

by suppressing terms of quadratic order and higher in c-1, where 0 is the angle

between v and f. In terms of the angles 6o, 61, and 0, the differential power becomes


dP q2p2 2
dP 22- h- (t)R(t)2 g(t) sin 60 sin g (t) sin 61 sin 0 (3.65)
dQ 2i2C s s
which is independent of the polar angle 9. The corresponding radiation pattern

is shown in Figure 3-7 and shows that power is radiated in all directions except

along the direction of motion. Notice that the "dipole-like" pattern is modified by

contributions arising from the expansion of the square roots in the angles 60 and 61.

That is

sin 2 2 V3 .62o
s1= c sin2 0 (,t0)cos0+ ((v/c)2cos20]. (3.66)
/I (v/ct)(t, to) cos0 L 2ct

These contributions are more significant at higher velocities. The quadrupole pat-

tern in Figure 3-7 is obviously overemphasized. By integration over the unit sphere,

the total power is found to be

q2t2 2
P(x, t) = h-(t)R(t)2fdo0{gj+(t) sin 6 Sin2 0 g+(t) in 6sin2 0} (3.67)
7c

which is equivalent to dE/dt where E is the total field energy. Both of the integrals

in (3.67) can be done analytically. Since both h- and g+ are proportional to 1/R,

the power does not decay with the radius x.










V











X


Figure 3-7: The radiation pattern given by (3.63) shows the characteristic dipole
pattern at lowest order. Keeping O(v/c) terms reveals the quadrupole pattern.
Higher order multiple patterns are generated by O(v2/c2) and higher terms.

For an electron whose charge distribution has a width corresponding to the

Bohr radius ao and has a velocity of ve 1 a.u. between the times to = 0 a.u. and

t1 = 1 a.u., the instantaneous power is P M 2 x 10-3 a.u. a 3 x 10-4 J/s at the

maximum of the peak from to. The power from the t1 peak is the same.

In order to put the previous results into perspective it is useful to make a

comparison with the Larmor result. The Larmor power

2q2
P(t) = (t)2 (3.68)

is the instantaneous power radiated by an accelerated point charge that is observed

in a reference frame where the velocity of the charge is significantly less than that

of light. The angular behavior of the emitted radiation may be determined by

examining the differential power

dP(t) q2
v(t)2 sin2 0 (3.69)
dQ 47Vc

which is the dipole radiation pattern. If the result of (3.65) is correct, then it should

reduce to the Larmor formula in the limit of the wavepacket width b going to zero









(point charge). Making use of the identity


6(x) lim e22 (3.70)
it+oo 11/7

where = 1//2b, the differential power in (3.65) becomes

dP q 2 [V (31)
d 4csin2 t- ) 5t- t)} (3.71)
a2

Again v/c < 1 was assumed. The term in square brackets has the dimensions of

acceleration. And so, (3.71) reduces to the Larmor result (3.69) for the stepwise

velocity v(t) = v6(t to)O(tl t). These results are presented in [88].















CHAPTER 4
CANONICAL STRUCTURE

The governing equation of quantum mechanics is the Schrodinger equation [7, 8].

In the minimal coupling prescription it is

S[-iV qA/c]2|
S+ V[ + pqn. (1.1)
2m

The dynamics of the scalar potential 4 and vector potential A are not described

by this linear equation. Specification of these potentials as well as the initial values

for the wavefunction T casts the Schr6dinger equation into a well-defined boundary

value problem that is also a well-defined initial value problem.

The governing equations of electrodynamics are Maxwell's equations [9]:

4w E B
V-E 47p Vx B 4- V B 0 VxE+B 0. (1.2)
c c c

The dynamics of the charge density p and current density J are not described by

these linear equations. Specification of the external sources as well as the initial

values for the electric and magnetic fields E and B -.l- i-vii,-; V E = 47p and

V B = 0 casts the Maxwell equations into a well-defined boundary value problem

that is also a well-defined initial value problem.

Each of these theories are significant in and of themselves. Given a particular

arrangement of sources throughout space-time and the initial values for E and B,

the Maxwell equations govern the dynamics of the resulting electromagnetic field.

Likewise, given a particular external field throughout space-time and the initial value

for ', the Schr6dinger equation governs the dynamics of the sources. However, notice

that the Maxwell equations do not say anything about the dynamics of the sources

and the Schrodinger equation does not say anything about the electrodynamics.









It is possible to couple the linear Maxwell and Schr6dinger equations. The

resulting nonlinear Maxwell-Schr6dinger theory accounts for the dynamics of the

charges and the electromagnetic field as well as their mutual interaction. For exam-

ple, given an initial source and its corresponding Coulomb field, a wavefunction and

electromagnetic field are generated. The electromagnetic field has its own dynamics

and acts back upon the wavefunction. This in turn causes different fields to be gen-

erated. It will be demonstrated that these coupled nonlinear Maxwell-Schr6dinger

equations can be cast into a well-defined initial value problem and solved in an

efficient numerical manner.

4.1 Lagrangian Electrodynamics

Consider the Maxwell Lagrangian density

[-Aa/c- V4]2- [Vx A]2 .J A
Livax = P+ -- (4.1)
87 c

with external sources p and J. Variation of this Lagrangian leads to the governing

equations of electrodynamics, i.e.,


C-2 C C
2 (1.4) 47
VV2I4
c
These Maxwell equations (in terms of the potentials) do not form a well-defined

initial value problem. But, by choosing a particular gauge they can be turned into

one. In other words, these equations are ill-posed as they stand. However, they do

enjoy both Lorentz and gauge invariance as does the Lagrangian (4.1).

4.1.1 Choosing a Gauge

Working in a particular gauge can be organized into the following hierarchy:

1. At the solution level, a gauge generator F can be chosen so that a gauge trans-
formation of the solutions, i.e., o -> P' = F/c and A -- A' A + VF, maps
them to new solutions that satisfy the gauge condition.









2. At the equation level, the set consisting of (1.4) together with a gauge constraint
has only solutions that satisfy the gauge condition.

3. At the Lagrangian level, a gauge fixing term can be added to (4.1) so that the
resulting Euler-Lagrange equations automatically include the gauge constraint.

4.1.2 The Lorenz and Coulomb Gauges

The first two tiers can be elaborated on as follows. With a gauge function F

-.i i-fvi,-; V2F F/c2 = -[/c+ V A] a solution A" (<, A) of the potential

equations (1.4) can be mapped to the Lorenz gauge solution Aiorenz according to

the gauge transformation:


4 -- (Lorenz = F/C A ALorenz A+ VF. (4.2)


Alternatively, adding the gauge constraint 4/c + V A =0 to (1.4) leads to the

Lorenz gauge equations of motion:

A 4w
V2A V2 2 47p. (4.3)
c2 c c

With p and J specified throughout space-time, the Lorenz gauge equations of mo-

tion are well-defined once the initial values for A, A, 4, and 4 are known. There

is some symmetry left in the solutions to these equations. Namely, the residual

gauge freedom left in the homogeneous equation V2F F//2 = 0 allows for gauge

transformations on the solutions such that the new solutions do not leave the Lorenz

gauge. However, these gauge transformed solutions do correspond to different initial

conditions. Note that the Lorenz gauge enjoys relativistic or Lorentz invariance. It

will be shown, that the Lorenz gauge is the most appropriate gauge for dynamics.

With another gauge function G -.I i-f i-n-i; V2G =-VA a solution A = (4, A)

of the potential equations (1.4) can be mapped to the Coulomb gauge solution

Acoulomb according to:


A ACoulomb = A + VG.


( -# (Coulomb = ( G/C


(4.4)









Alternatively, adding the gauge constraint V A = 0 to (1.4) leads to the Coulomb

gauge equations of motion:

A 4x V,
V2A V = -47p. (4.5)
C2 C C

Again with p and J specified throughout space-time, the Coulomb gauge equations

of motion are well-defined once the initial values for A, A, 4, and 4 are known. As

before, there remains a symmetry or residual gauge freedom from the homogeneous

equation V2G = 0. Note that in the Coulomb gauge Gauss's law reduces to V2

-47p. Inverting this equation specifies D in terms of p. That is D = (1/V2)[-47p].

The scalar potential can now be totally removed from the theory by substitution

of this Green's function integral. This may be done at the expense of Lorentz

invariance. In practice, where the equations are to be expanded in a basis of s-

gaussians, either transverse basis functions would have to be used or the transverse

fields would have to be generated from a standard basis. The former case would

require a major revision of most existing integral codes, which are in direct space,

while the latter would require the instantaneous transverse projection pfb ab

Oaab/v2 (see Appendix B) This operation, which is over all space, is difficult to

describe in terms of a local set of basis functions.

Lastly, for the third tier, consider the Lagrangian density (4.1) together with a

gauge fixing term for the Lorenz gauge, i.e.,

S[-,/c + V A]2
!2Max -2Max -
8M (1.18)
[-A/c- V(]2 [V x A]2 J- A [,/c + V A]2
-p +pI+
87 c 87
The resulting Euler-Lagrange equations obtained from max are identical to the

Lorenz gauge wave equations in (4.3) which are equivalent to the general potential

equations (1.4) together with the constraint E/c + V A = 0.









MCax LMax


d Mx_ 8tMax 0 d OMx_ 8tMax
dt ~ do = dt o ~
V A 0 gauge invariant
well-posed IVP add ill-posed IVP
constraint


unique .
solution gauge
transformation


solutions

$\


L
Max



dt o .~
>- V A + '/c 0
add well-posed IVP
constraint


gauge
transformation


unique
solution


Maxwell's equations
well-posed IVP

Figure 4-1: A limited but relevant portion of the gauge story in the Lagrangian
formalism is organized in this picture. The middle column (i.e., the column below
Max) enjoys full gauge freedom. The far left (Coulomb gauge) and far right (Lorenz
gauge) columns have limited gauge freedom. That is, there are a limited class of
gauge transformations that can be made on the solutions such that they remain in
the same gauge. This symmetry is due to the residual gauge freedom. Note that
these solutions correspond to different initial conditions within the gauge. Also note
that the Euler-Lagrange equations together with a particular gauge constraint are
equivalent to the Euler-Lagrange equations derived from that particular gauge fixed
Lagrangian.


There are many other known gauges, the choice of which is arbitrary. All

choices of gauge lead to the same pi l,-i, .,lly observable electromagnetic fields E and


B. Together with the definitions E


A/c V( and B = V x A, the Lorenz and


Coulomb gauge equations of motion as well as the general potential equations (1.4)

imply Maxwell's equations (1.2). A diagram of this gauge story in the Lagrange

formulation is presented in Figure 4-1.

4.2 Hamiltonian Electrodynamics

In the Hamiltonian prescription, the momentum conjugate to A with respect

to the Maxwell Lagrangian (4.1) is


SMax
9A


1
-[A/c+ V4].
47c


(4.6)







60

The momentum conjugate to D is identically zero, i.e.,

SE a = 0. (4.7)


A Hamiltonian density can still be defined as the time-time component of the

Maxwell stress-energy tensor T'x {OMax/O/(a)}O'ac gt3Max- It is

x ax -n.A +x [-4cH]2+2 [Vx A].2 J+ A
87 c
(4.8)

and the resulting equations of motion are:

O -Max O'-M~2ax V[V A] V2A J
A = 47c2n cV(I -n + cV
0nH A 47 c
(4.9)
OM ax O ax
=- 9 0 -9- (9+cVH1.


Since the momentum 6 defined in (4.7) is identically zero, so is its time derivative

6 and gradient VO. Notice that these Hamilton equations form a well-posed initial

value problem. The machinery inherent in the Hamiltonian formalism automatically

adds a momentum and automatically adds the additional equation of constraint

4 = 0. It turns out that this extra equation fixes a particular gauge where 4 = 0.

This gauge can always be fixed by a gauge transformation whose generator satisfies

F/c = D. The residual gauge freedom left in the homogeneous equation F = 0

does allow for a gauge transformation on the solutions to (4.9). These new gauge

transformed solutions do not leave the 4 = 0 gauge, but do correspond to a different

initial value problem within this gauge. In other words, they are solutions to (4.9)

with different initial values. Pay careful attention to the fact that these Hamilton

equations of motion form a well-posed initial value problem even though a gauge

fixed Lagrangian was not knowingly used. The Hamiltonian formalism automatically

added the extra equation 4 = 0.







61

4.2.1 Hamiltonian Formulation of the Lorenz Gauge

Rather than fixing the Coulomb gauge at the equation level it may be bene-

ficial to work in a more general theory where a gauge is chosen at the Lagrangian

level and retains all of the 4-potential, is Lorentz invariant, and does not require

any instantaneous or nonlocal operations. To this end, consider the Lorenz gauge

Lagrangian density from (1.18), i.e.,

L [-A/c V4]2 [V x A]2 J. A [V/c + 7V A]2


tvMax


cf 87
c 8i


(I.Io)


It will be shown that the equations of motion derived from Lax are well-defined

because of the addition of the last term in this expression. It turns out that this

term is known in the literature [24, 68] and is a gauge fixing term for the Lorenz

gauge. From (1.18), the momentum conjugate to A is


rL 1
S -ax 1 [A/c+ V]
9A 47rc


(4.10)


and the momentum conjugate to ( is


SMax
9 ^


1
-- [4/c + V A].
47c


(4.11)


With these moment and coordinates, electrodynamics is given a symplectic struc-

ture. The Hamiltonian density is

jL [-47cir]2 + [V x A]2 [47rc 2 ,, ,- J.A 1~


tv, z 'L t- Av i t/L -' IJJz.IZ}
C


and the resulting equations of motion are:


A = 47c2H cV7I


V[V. A] -V2A J IV
47r c


(4.13)


4 = -47c2( cV A


-6 p + cV n.


S-Max









These equations, which are a generalization of (4.3), together with the initial values

for A, H, 4, and 6 form a well-posed initial value problem. The residual gauge

freedom resulting from the homogeneous equation OF = 0 does allow for a gauge

transformation on the solutions to (4.13). These new gauge transformed solutions

do not leave the Lorenz gauge, but do correspond to a different initial value problem

within the Lorenz gauge. In other words, they are solutions to (4.13) with different

initial values.

Notice that a relationship exists between the momentum 6 and the gauge func-

tion F leading to the Lorenz gauge. That is, from = -[(I/c + V A]/4rc and

F/c2 V2F = ,/c + V A notice that O OF/47c. So the D'Alembertian of

the gauge function F acts a generalized coordinate in this phase space. It is the

momentum conjugate to the scalar potential 4.

In matrix form, the dynamical equations in (4.13) are

0 0 -1 0 A Vx [VxA]/4 -J/c + cVO

0 0 0 -1 # p + cV
(4.14)
1 0 0 0 I 47rc2n cV)

0 1 0 0 -47c20 cV A

where 1 is the 3 x 3 identity matrix. Notice that (4.14) is of the Hamiltonian form


wu) = aH/r0. (4.15)


More specifically labfb = -H/i9a, where rb is a column matrix of the generalized

positions and moment, i.e.,

Ak


b (x, t) (4.16)
IIk
19









where k = 1, 2, 3. The antisymmetric matrix Wcab is the (canonical) symplectic form

associated with the phase space of electrodynamics in the Lorenz gauge. By substi-

tution, these first order Hamiltonian equations of motion can be shown to be equiv-

alent to the second order Lorenz gauge equations D4 = -47p and OA = -47J/c.

Together with the definition of the electric and magnetic fields, (4.13) imply


V E 47p + 47r V x B 4J E 4cVO
c c
(4.17)
B
V-B 0 VxE+--0.
c

These equations are not equivalent to Maxwell's equations unless 6(x, t) remains

constant in space-time throughout the dynamics. In order to analyze this question,

the dynamics of the sources must be considered. It should be noticed that the

inhomogeneous equations in (4.17) imply

6 1
DO V2 _[P + V. J]. (4.18)
S 2 2 C2

If the matter theory is such that the equation of continuity p = -V J is satisfied,

then DO 0. So if G(t = 0) -= (t = 0) = 0, then 0(t) = 0 at all times t. In other

words, if the sources of charge and current satisfy the equation of continuity, then

the dynamical theory arising from the Lagrangian (1.18) is the Maxwell theory of

electrodynamics.

Note that while (4.9) and (4.13) do not enjoy the full gauge symmetry as do the

general potential equations (1.4), this does not mean that the observables resulting

from (4.9) or (4.13) are not gauge invariant. Any observable that is calculated will

be invariant to the choice of gauge generator. Moreover, once the solutions to these

well-defined equations are constructed, these solutions belong to the many solutions

of (1.4). This family of solutions is the most general solutions of the potential form

of Maxwell's equations. In fact, gauge transformations can even be made from one







64

particular gauge to another [89]. A diagram depicting the relevant gauge story in

the Hamiltonian formulation is presented in Figure 4-2. Notice that there is no

Hamiltonian theory that enjoys the full gauge symmetry of (1.4). The Hamiltonian

H-Max in the far right column is obtained by a Legendre transformation of the gauge

invariant Lagrangian mMax in (4.1). However, the Hamiltonian dynamics stemming

from the gauge invariant Max is not gauge invariant, but rather occurs in the gauge

where 4 = 0.















Max Max M


d OtM.a OtM.x 0 d OtM.x tM.x- 0 d tMax tmMax 0
dt 8 a~ dt 0a ai dt a as
V A 0 gauge invariant > V A + 4/c 0
well-posed IVP add ill-posed IVP add well-posed IVP
consraconstra


gauge
transformation


many
solutions gauge
I transformation


Maxwell's euations
well-posed IVP


O7-CMax
well-posed IVP



MFaC Lax rC
Max 0 Max


unique
solution
4!


gauge
transformation


aIL
well-posed IVP


%H-L MLax
Max -a Max


unique
solution
A


well-posed IVP



R-Max -Lx a Max


Figure 4-2: The Hamiltonian formulation of the gauge story is organized in this picture with respect to the previous Lagrangian
formulation. Figure 4-1 is depicted in the box with dotted borders. It can now be seen how the Coulomb and Lorenz gauges
connect in both formalisms.


unique
solution


$









4.2.2 Poisson Bracket for Electrodynamics

The phase space that carries the associated dynamics is naturally endowed

with a Poisson bracket {., .} (recall Chapter 2). This may be seen by considering

the variation of along the dynamics AH = (a/9rj)r. That is


A() b (d/d) = ( lb)b = ( rb)I(H/9r) = {(, H}, (4.19)

where rj are the generalized coordinates. In general, the Poisson bracket of the

dynamical variable F with the dynamical variable G is

SF/0A 0 0 -1 0 G/9A
9F/9, o o o -1 9G/9^
{F,G} 0 0 0 1 (4.20)
OF/aI 1 0 0 0 OG/ac

OFO / 0 1 0 0 G/OO

Since the symplectic form w is canonical its inverse is trivial, i.e., 1 = T = -.

Also notice that w2 = -1, Tw = 1, and det w = 1.

4.3 Hamiltonian Electrodynamics and Wave Mechanics in Complex
Phase Space

Consider the matter theory associated with the Schrodinger Lagrangian (h = 1)


sch [i qA/c]W* [-iV qA/c]> V*- q (1.17)
2m
where T is the wavefunction for a single electron, V = qq/|xl is the static Coulomb

potential energy of a proton, and ((, A) are the electron's scalar and vector poten-

tials. Notice that this Lagrangian is already written in phase space. The momentum

conjugate to the wavefunction T is iI*. Together with the previous Maxwell La-

grangian, the coupled nonlinear dynmical theory arising from the Lagrangians

1 IrAr [-4rcHl]2+ [Vx A]2 v84
LMax A H A] cV H} (4.21)
2 87












S{[iv qA/c]T* [-iV qA/c]W
LSch 2 ++ (4.22)
(4.22)


1
gauge [ [( O ] {-27rc22 cOV A}
2


(4.23)


yields the following equations of motion:


A = 47rc2 cV

4 -47c2( cV A
S[-iV qA/c]2
i2 + VT + q 2mr


-n

-e

-i J*


V[V. A]- V2A J
47 C
cVO

p + cV. n (4.24)
[iV qA/c]2*
2 + V* + q,"*.
2m


Surface terms of the form (d/dt){pq/2} have been added in the above Lagrangians

in order to symmetrize them, i.e., L = pq H (d/dt){pq/2} becomes L = r/," -

pq]/2 H. This can always be done since the action I = f Ldt = f[L + (d/dt)g]dt

is invariant to the addition of a pure surface term to the Lagrangian. Note that

the Schr6dinger wavefunctions T and I* are complex-valued while the remaining

electromagnetic variables are all real-valued. These dynamical equations may be

put into matrix form as

/ 0 0 0 0 0 / \ \ / f-iV nA/ lrT/2 m -- VT -- on( \


0 -i 0 0 0 0

0 0 0 0 -1 0

0 0 0 0 0 -1

0 0 1 0 0 0


A



H


[iV qA/c]2 */2m + VT* + qT*

V x [Vx A]/47 J/c cVe

p +cV H

47c21 cV<


0 0 1 0 0 -4rc( cV A
(4.25)

the symplectic form is canonical. The electromagnetic sector of it is identi-

(4.14). These dynamical equations define the coupled Maxwell-Schr6dinger


0


where

cal to









theory. This theory is well-defined and closed. In other words, the dynamics of

the charges, currents, and fields are all specified as well as their mutual interaction.

Given initial values for I, I*, A, H, 4, and 6 determines their coupled dynamics

throughout space-time.

With the dynamics of the charges defined, the problem in (4.17) can now be

addressed. The Schr6dinger equation in (4.24) implies the continuity equation


(d/dt)qT* = -V q{j*[-iV qA/c] + [iV qA/c]*}/2m (4.26)

which may be written more compactly as p = -V J. From the definition of the

momentum 6 in (4.11) and the wave equations E< = -47p and OA = -47J/c,

notice that

-1 1 1
D0 I[(d/dt)OD/c+V A] I[(d/dt)4p/c+V- 4J/c] -2 [p+V- J] 0
4i7c 4i7c C2
(4.27)

by appealing to (4.26). So if 6(t = 0) = G(t = 0) = 0, then the electrodynamics
- .,,- in the Lorenz gauge for all time since the only solution of =- 0 with

G(t = 0) -= (t = 0) 0 is 0(t) = 0.

It is worth mentioning that if G(t = 0) = 0 for all time, then the electron-

electron self interaction makes no contribution to the Schr6dinger energy. This

is true since the self interaction term q44*I in the above Schr6dinger Lagrangian

cancels exactly with -cV-HI in the Maxwell Lagrangian. The cancellation requires

a partial integration of -cV4 H to clV H followed by a substitution of 0 =

p + cV H from (t = 0) = 0 in (4.24). However, there is still a contribution from

the self-energy arising in the Maxwell energy of the Coulombic field.









4.4 Hamiltonian Electrodynamics and Wave Mechanics in Real Phase
Space

The dynamical equations (5.16) are mixed, real and complex. For consistency
these equations are put into real form with the Lagrangian densities:

1 In A[-4 ]2 + [ V x A]-2 _4C77l _
Max [H A HA1 4- cV } 8 (4.28)
2 8i



1 +{[VQ qAP/c]2 + [-VP + qAQ/c]2}/4
=Sch -[PQ-PQ]- (4.29)
[2 +V[Q2 +P2]/2 + q'[Q2 P2]/2



gauge [e6 6O] {-27i 2 C- ce A} (4.30)

The functions P and Q are related to the real and imaginary parts of I and I*
according to T = [Q + iP]/v2 and I* = [Q iP]/v2. The equations of motion

that are associated with these Lagrangians are:

4 7rc2H cV4 7V[V A] -V2A J cV
47 c
(4.31a)

S= -47rc2 cV A + = p + cV -


-V2P+ qV (AQ)/c+ qA. VQ/c+ q2A2P/c2 VP
2m
(4.31b)
V2Q + qV (AP)/c+ qVP A/c q2A2Q/2 V .
2m









These dynamical equations may be put into matrix form as

0 0 0 -1 0 0 A

0 0 0 0 -1 0

0 0 0 0 0 -1 Q

1 0 0 0 0 n

0 1 0 0 0 0 8

0 0 1 0 0 0 P

V x [V x A]/47 J/c + cVO

p +cV H

-[V2Q + qV (AP)/c+ qVP A/c q2A2Q/c2]/2m VQ + qQ

47rc2n cV<

-47c2( cV A

[-V2P + qV (AQ)/c + qA VQ/c+ q/cq2A2p ] /2 + VP + qP

(4.32)

where the symplectic form is again canonical. Note that the equation of continuity

p = -V J still holds with the real charge and current densities


q[Q2+ p2]/2 J {QVP- PVQ qQAQ/c qPAP/c}. (4.33)
2m

4.5 The Coulomb Reference by Canonical Transformation

As was mentioned previously the numerical implementation of the theory can

be made to converge more quickly if the basis is chosen judiciously. Recall that

the electromagnetic field generated by any charge contains a Coulombic contribu-

tion. This monopole term accounts for a large portion of the local electromagnetic

field surrounding the charge. It would be advantageous to not describe this large









contribution in terms of the basis but rather to calculate it analytically. The re-
maining smaller portion of the radiative or dynamical electromagnetic field can then

be described in terms of the basis.

To this end, notice that the scalar potential = c + (( c) (c +(D may

be split into a Coulombic portion -.I i-fviir-; V2fc = -4rp that can be calculated

analytically and a remainder (D regardless of the choice of gauge. The Coulombic

potential is not itself a dynamical variable but depends on the dynamical variables

Q and P. That is
The dynamical portion (D is a generalized coordinate and is represented in the

basis. Similarly, the momentum conjugate to A may be split into a Coulombic and

dynamical piece according to

a0Mx VCc 1
H 4 Hc + HD = rc+ [A/c + V4D]. (4.34)
O9A 47c 47c

Like iD, the dynamical portion HD is a generalized coordinate and is represented

in the basis.

4.5.1 Symplectic Transformation to the Coulomb Reference

The transformation to these new coordinates, i.e., (D and 11D, is obtained by

the canonical or symplectic transformation

A A(A) A
(D #i( D, P) D Dc(Q, P)

Q O(Q) Q
T: (4.35)
n n(n, Q, P) nc(Q, P)

e o0() e

P P(P) P

where 4 (D and l 1D. The variables Q, P, A, and O are unchanged by T.

Since both (c and He are complicated functions of Q and P, the inversion of T may









be quite involved. However, it will be shown that the inverse of T does exist. In fact

both the T and T-1 are differentiable mappings on symplectic manifolds. Therefore

the canonical transformation is a symplectic diffeomorphism or symplectomorphism

[29].

The theory of restricted (i.e., explicitly time-independent) canonical transfor-

mations [27, 28] gives the general prescription for the transformation of the old

Hamilton equations (4.32) to the new Hamilton equations in terms of T (and TT)

only. In symbols, that is

-1H OH
rj = W -- A 1= H (4.36)

where the new Hamiltonian H is equivalent to the old Hamiltonian H expressed in

terms of the new variables 17. (For simplicity H will be written as H from this point

forward.) To this end, consider the time derivative of the new column matrix


rli i.= or r= Trj. (4.37)
Aqj

Substituting r) from (4.36) results in

IH OH
%i =Ti- or r= TW-1l (4.38)
a3 k akr)

Lastly the column matrix OH/9r] can be written as

OH OH 0l =TOH OH TOH
Sor = T (4.39)
0%k 0%1 9rk 9H 9r] 9

so that the new equations of motion (4.38) become

STOH iOH IOH
= oTakjT -Wor Tw-1Tw_ a -1 (4.40)
3 Nkl A Oi









This canonical transformation on the equations of motion leaves only the com-
putation of C-1 = Tw-1TT since the Hamiltonian automatically becomes

[-47rc{I + Hc(Q, P)}]2 [+ V x A]2 [47rcl2 q[2 P2
R = 8{+ q[) + xc(Q, P)]
8ir 2
cV[E + Oc(Q, P)] [H + Hc(Q, P)] cOV A

[VQ + qAP/c]2 + [-VP AQ/c]2 + 2 + P2
4m 2
(4.41)

in terms of the new coordinates. However, the inversion of w is not simple in practice.

It turns out that the equations of motion (4.36) are most practically written as

OH H _H
-= = (T- )T T-1 not C = -H = Tw- TT H (4.42)

where the inverse transformation T-1 is the transformation of the inverse il1.1 i'ii,-:.
i.e., Ti-1 9]i/9rj. It will be shown that det T / 0 so the mapping is well-defined.
These equations of motion are of the desired form because they involve w and not
W-1. That w-1 is undesirable is seen by going to the basis. In the basis, the canonical
symplectic form becomes


0 ( (4.43)
1 Q a(P\Q) 0

which is not easily inverted. As a result it is simpler to compute (T-1)TcwT-1 than
Tc-1TT even though T-1 is needed in the former case. It will be shown that the

explicit evaluation of T-1 is not necessary.

To continue with the transformed equations of motion in (4.42), which only
require w, the mapping (T-1)T : 9/9r -4 9/19) must first be set up. The transposed
inverse transformation (T-1)T is defined on the vector fields themselves according











a/aA
ato








OA/OA 0 0 0 0 0 O/OA
o 0a/a, o o o0 0o /a
o a'/aQ OQ/OQ aHn/a 0 0o O/Q
o o 0 an/an 0 0 a/an
0 0 0 0 0e/0e 0 a/ae
o o o o aoe/a o a/aoe
0 o o/8P 0 an/8P 0 9P/9P ) 9/9P
(4.44)

Notice that
O(A, (b Q, H, o, P)
det(T-1) det T- (det T)-1 Q, H,
0(A, (, Q, H, n P) (4.45)
= (9A/9A)(9^/9 )(9Q/9Q) (an/an)(9G/9O)(9P/9P)= 1

so that the transformation is canonical and symplectic or area preserving. In other
words, the new infinitesimal volume element drf is related to the old infinitesimal
volume element drj by
dF = det T dr = drj (4.46)

since the determinant of the Jacobian is unity. Thus, the volume element of phase
space is the same before and after the transformation. It is a canonical invariant.








With (T-1)T the similarity transformation of the canonical symplectic form in
(4.32) is


Co = (T-1)TWT-1


(m Q>
QV cG)


(4.47)


where


(9A/9A)(-1)(Hn/HfI)
>= o0
0


(an/aH)(1)(aA/aA)
0
(9Hn/9P)(9A/9A)


0 -(9A/9A)(nIH/P)
(a9/a)(- 1)(a9/a9) 0
-(9^/9Q)(9e/e8) (9Q/9Q)(-1)((P/8P)
(4.48)


0
(a9/ao) (l)(a/a>)
0o


0 0
0 0
(an/aQ)(aA/aA) 0


0
(a9/a9) (a9/aQ)
(OP/9P) (1) (Q/Q) )
(4.49)


-(aA/A)(HIan/Q) )
0
0 )


(4.50)


( 0 0 0
G= 0 0 (9e/9e)(9^/9P) 0 (4.51)
0 -(84/l P)(9e6/9) 0
The factors of 1 and -1 are explicitly written in Q> and Qv to bring out their
similarity to the canonical symplectic form in (4.32). After computing the derivates


tiV


~2n/r I









in w it can be shown that w equals


0

0

anc(Q, P)/9Q
1



onc(Q, P)/9P


o -anc(Q, P)/Q

o o

o o

o o

1 aOc(Q,P)/aQ

o 1


o

-1

-a(c(Q, P) /9Q

0

0

-O(c(Q, P)/OP


-anc(Q, P)/aP

0
-1

0

ODc (Q, P)/O9
0 (
(4.52)


with


q Q(x', t)2 + (x', t) 2
c(x, t) fy
2 |~x-x|


(4.53)


and
q Q(x', t)2 +P(x', )2
Ic(x, t) vfv dx3. (4.54)
87c |x x'|
And so the new symplectic form contains extra elements that are not present in the

canonical w. These extra elements add additional time-dependent couplings to the

theory. As before, the associated phase space is naturally endowed with the Poisson

bracket


{F, G} =- (aF/ar)Tw"O- (G/9ar).


(4.55)









The transformed equations of motion with symplectic form (4.52) may be writ-
ten in full as:

oanc h OH V x [V x A]
Q P A cV
8Q aP aA 4x


. A
9Q


OH

aH
OQ


Qc
00d


A
0DC Oc -/
-D+-Q + -P
aQ OP
9Hoc 9 - A+Q -
OP OP


aH

OH

aH
aH
aP


QP-PVQ QAQ -PAP1
2mcI c c i
Q2 + P2
q- + cV [n + Tic]
2
-V2Q qV (AP)/c qVP A/c + q2A2Q/C2
2m

+VQ+q[ + c]Q

4 2 -co A aQ



47-c2[f + cl cV[] + C]

-47rc2 cV A

-V2P + qV (AQ)/c + qA VQ/c + q2A2P/c2


+VP + q[ + c]P
anH
+ {47c2[H + IC] cV[ + +c]} /5
bP

2 2 cV. [ + H o1]

(4.56)

where Ic nc(Q, P) and bc -= c(Q, P). The forces appearing on the right
hand side of these equations have become more complicated, especially those in the
Schr6dinger equations. There are new nonlinear terms. However, it is possible to
substitute these equations among themselves in order to simplify them. Notice that
parts of the OH/9O and OH/OH equations appear in the forces of the Schr6dinger









equations. Substitution of 9H/9' and OH/IH into the Schr6dinger equations re-

sults in the following simplified equations:

anc an Vx [Vx A]
Q- H- P -+cV_
aQ OP 47
q{QVP -PVQ QAQ -PAP- }
2mc c c
S QO2 P2
q = 2 +cV. [I + c]

S V2Q + qV (AP)/c + qVP q A/c q2A /c2 C2
-P=2 + VQ
2m
+ q[b +c]Q

A = 47cc2[f + H IC] cV[) + (DC]

S+ Q + P =-4c2 cV A
aQ 9P
V2 + qV (AQ)/c qA VQ/c +q2AP2 C2
Q= 7 +VP
2m
+q[4b+c]P.

(4.57)

The generalized forces appearing on the right hand side are now very similar to the

forces in (4.31). In fact, the equations of motion (4.57) can be further simplified as:

-[nH + ri (Q,P)] -= H/9A A = H/H
= 9H/94 4 c+ c(Q, P) -= H/9Q (4.58)
-P -= H/9Q Q = H/aP,

where the tildes were omitted to show the resemblance between (4.58) and (4.31).

4.5.2 The Coulomb Reference by Change of Variable

It can be shown that the new equations of motion wrj = H/9ri, which were

obtained by a symplectic transformation in phase space, may also be obtained by a

change of variable in the Lagrangians (4.28)-(4.30). The new Lagrangian density is:








Change of Variables
(p, q) >






wr dlr


L ,









c ii) aH/ar,


Canonical Transformation

Figure 4-3: Commutivity diagram representing the change of coordinates (q, p) to
(p, q) at both the Lagrangian and equation of motion levels.




S({ I+ I[-4 c(H + i H)]2 + [V x A]2}/8
Mx -2[(n+n)- A-(H+ c)-(4.5A]-9)
S-cV [b + cD] [H + HC]
(4.59)


sc
Sch= -[PQ
2


P] { {[VQ + qAP/c]2 + [- + qAQ/c]2/4 (4.60)
P]V q( c 2 (4.60)
+[V+q(b+DC)]{Q 2 p2


1
gauge =- [(( i + ic) 6( + (c)] {-2c22 cV A}. (4.61)

That the transformation to the Coulomb reference holds at both Lagrangian and

equation of motion level demonstrates the commutivity of the diagram in Figure
4-3.

4.6 Electron Spin in the Pauli Theory

The electron field used so far in the nonrelativistic Schridinger theory is a field

of spin zero, i.e., a scalar field. It is a simple generalization of the theory to add in

the electron's spin. The electron field would then be a two component spinor field,








i.e., a spin-1/2 field, and would be of the form

Ip (x, t) ( .I t)0 (4.62)

The first component Ti is spin up and the second component 41 is spin down. The
dynamics of Tp is governed by the Pauli equation [30]

iip [-i qA/c]2P + Vlp + qP _p o- [V x A]p (4.63)
2m 2mc

which is the nonrelativistic limit of the Dirac equation

i4D /3= pc2 D + ca [-iV qA/c]D + q D (4.64)

in terms of the four component spinor oD, where the 3 and a matrices are


S(01) (4.65)

and
0 0 -i 1 \
-= a ) J) az I = (4.66)
1 0 1 0 0 -1
Notice that taking the nonrelativistic limit of the Dirac equation involves the elim-
ination of the two component positron field from TD. Also note that the current
density associated with the Pauli theory [90] is different from that in the Schrodinger
theory (see (1.14)). It is

Jp = q [-iV qA/c]p + 1p[iV qA/c]tp + V x [t1or'p]}/2m, (4.67)

where 1t = (4* I*) is the adjoint of Ip. This can be derived by taking the
nonrelativistic limit of the Dirac current density. The last term in (4.67) is only
present in the Pauli current. This term does not effect the continuity equation
p = -V J since V V x [Tcr4p] = 0.









4.7 Proton Dynamics

In the theory set up so far, the matter dynamics was entirely described by

the electronic wavefunction T. The proton had no dynamics whatsoever. Only

the electrostatic scalar potential q = q/|x| of the structureless proton of charge

q entered so as to bind the electron in the hydrogen atom. A first step in the

direction of atomic and molecular collisions requires the dynamics of the proton as

well (and eventually a few other particles). Suppose the proton is described by its

own wavefunction C and Lagrangian density

h ijq2Q [iV qA/c]* *. [-iV qA/c]
ch = in-* -4. *Q, (4.68)
2mq

where (b, A) are the scalar and vector potentials arising from the charge and current
densities

p = qT*T + qfQ* (4.69)


J q{(*[-iV qA/c]i + '[iV qA/c]i*}/2m,
(4.70)
+ q{(*[-iV qA/c]QO + [iV qA/c]Q*}/2mq.

These densities are just the sum of the individual electronic and proton densities.

The proton density is not a delta function. Thus, the proton wavefunction is not a

delta function either. Rather it is described by a wavepacket and has some structure.

With (4.68) the total Lagrangian is

.1 [ [-47rcH]2 + [Vx A]2
Max = -A H- ..A]- j I|cV|H (4.71)
2 7[iV



Lch q [iV qA/c]* [-iV qA/c]'
2 2mq +n











ch i [*Q 2*2] {[iV qA/c]* [-iV qA/c] + qQ*Q (4.73)
L4ch [* O*Q] + q^* (4.73)
2 2qm,



gauge [O6 O(I] {-27rc22 cOV A}. (4.74)

Notice that the electron Lagrangian (4.72) does not explicitly contain the static pro-

ton potential energy V = qq/|xl as did the previous Schrodinger Lagrangian (1.17).

The two matter fields are coupled entirely through electrodynamics. That is, the

electron-proton interaction is mediated by the electrodynamics. The Coulombic

potential is included implicitly in q(I4*~ and q(I2*Q in the above matter Hamil-

tonians. In other words, the scalar potential ( contains (in any gauge) a Coulomb

piece of the form

c (x, t) = + (c ) = Jv x' d
x Xx'
q*(x', )(x', t) + *(x', t)(x, t) d .
v |_X x'| J x x '
(4.75)

With this potential, the q4I*J term in the electron Hamiltonian contains the

electron-proton attraction as well as electron-electron self interaction. Similarly

the qQ2*Q in the proton Hamiltonian contains the electron-proton attraction and

proton-proton self interaction.

The self-energies that are computed from the aforementioned self interactions

are finite because T and 2 are square integrable functions. That is

Eit = J (x, t))c(x, t)d3x = fd3xfdx(xt)(xt)' < (4.76)
x x'|

for both the cross terms (electron-proton attraction) and the direct terms (electron-

electron and proton-proton repulsion). Note that in the relativistic quantum theory

the direct terms are infinite and there are infinitely many Coulomb states of the









bare problem to sum over [40]. These infinities do not arise in the semiclassical

theory presented in this dissertation. While the self interactions do appear in the

above matter Hamiltonians, the resulting self-energies are finite and moreover do

not even contribute to the electron or proton portions of the energy. This is due to

-cV4 IH in the above Maxwell Hamiltonian. After a partial integration this term

becomes c4V H. Substitution of -O = p + cV H = 0 from (4.31) turns c4V H

into -pK, which cancels +pP in the electron and proton energies. However, the self

interactions do remain in the Coulomb energy E2/87 of the electromagnetic field.

Note that the self interactions do appear in the Hamiltonians and therefore do make

a contribution to the overall dynamics.

It should be mentioned that this theory of electron-proton dynamics can be

applied to electron-positron dynamics as well. While there is a 2000-fold difference

in mass between the proton and the positron, the two theories are otherwise identical.

In either case, the theory may be rich enough to capture bound states of hydrogen

or positronium.















CHAPTER 5
NUMERICAL IMPLEMENTATION

The formal theory of Maxwell-Schrodinger dynamics was constructed in the

previous chapter. In particular, the coupled and nonlinear Maxwell-Schrodinger

equations
S[P qA/c]2i
2= + VW + q#, (5.1)
2m


VA -47
V2A VL -V. A +- 4-J (5.2a)

V24 A -4xp (5.2b)
c

were recognized to be ill-posed unless an extra equation of constraint is added to

them. Using the Hamiltonain approach to dynamics, this extra equation was auto-

matically generated by adding a Lorenz gauge fixing term at the Lagrangian level.

It was emphasized in Chapter 4 that the resulting Hamiltonian system of differential

equations, which are of first order in time, form a well-defined initial value problem.

That is, the Maxwell-Schrodinger dynamics are known in principle once the initial

values are specified for each of the dynamical variables.

The details of converting the formal mathematics of Chapter 4 to a form that

can be practically implemented in a computer are presented in this chapter. The

Hamiltonian system of partial differential equations will be reduced to a Hamilto-

nian system of ordinary differential equations in time by introducing a spatial basis

for each of the dynamical variables. The resulting basis equations are coded in a

FORTRAN 90 computer program. With this program, various pictures are made

to depict the dynamics of the hydrogen atom interacting with the electromagnetic

field.









5.1 Maxwell-Schr6dinger Theory in a Complex Basis

Each of the Maxwell-Schrodinger dynamical variables, which are themselves

fields, may be expanded into a complete basis of functions Gk according to

I(x, t) = EG(x) (t) I*(x, t) = ECG(x )" (t)

Ak(x, t) = EGC(x)akK(t) nk(x, t) = EG(x)k(t) (5.3)

D(x, t) = EZK G(x) (t) (x, t) = EcG(x)Oc(t),

where the index IC runs over the basis and the index k runs over 1, 2, 3 or x, y, z.

Any complete set of functions such as the oscillator eigenstates will suffice. In the

following work the set of gaussian functions of the form

G (x) G* (x) = Nk exp(- f [x rk]2) (5.4)

are used. These functions are centered on rk, normalized to unity by Nk, and are

real-valued. Additionally, they span L2 so that any square integrable function may

be represented in this basis. In principle the sums in (5.3) are to infinity. However,

a complete basis cannot be realized in practice. But for all practical purposes the

numerical results can be shown to converge to within an arbitrary accuracy in a finite

basis. In fact with a smart choice of basis, the numerical results may converge with

just a few terms. Here the basis coefficients, which are complex- and real-valued as

well as time-dependent, carry the dynamics.

The basis representation of the previous Lagrangians is


LMax = Ek I [(a/0amM)am (0/807mM)TmM]SMax HMax (5.5)



LSch E [(/ K)K (a/ '1 ch HSch (5.6)



Lgauge EY [(a/aC): (0/0k)Ok]Sgauge Hgauge (5.7)









with integrals


SMax -= f Ad3x


SSch Sv *qId3x


Sgauge f v'd x.


The calculus of variations leads to the following dynamical equations:


2 SMax OH
,lMO,,\ nA\r mM
SG'7TnA


02Sgauge OH



02iSSch OH
01, i" "


2 SMax
C(<,,\r^rnarM


OH
( l,,.\r


2 Sgauge



02/SSch
a9Jo-,,


OH



aH
9^j


or MmM,ngxin =- Vw,



or Nzcc = VoH



or iCzc = V, H



o- -~.,mmM = Va



or N O = V0,H


which are of the Hamiltonian form wr = H/dri. The summation convention is used

throughout. These equations may be written more compactly as


Ma = VH

N = VoH


MTr = VaH

NT = V7H


(5.15)


i~ V~HiC*yj* V9.,H


(5.8)


mMH
nM.


(5.9)




(5.10)



(5.11)



(5.12)


n H


(5.13)


(5.14)









and can be cast into matrix form as

iC 0 0 0 0 0 y H/a*

0 -iC* 0 0 0 0 aH/9a

0 0 0 0 -MT 0 a OH/9a
(5.16)
0 0 0 0 0 -NT H/90

0 0 M 0 0 0 a OH/0t7

0 0 0 N 0 0 0 9H/90

where the matrices M, N, and C and defined in (5.9)-(5.14). This symplectic form

almost has the canonical structure of (4.25). In a basis of rank N, the contractions

involving a and i run to 3N while the contractions involving the remaining dy-

namical variables run to N. This is because a and r are spatial vectors that have

(x, y, z)-components whereas the remaining dynamical variables are scalars.
With the choice of representation in (5.3) and the choice of basis in (5.4) all

approximations are specified. The equations of motion in (5.16) are the basis rep-

resentation of the coupled Maxwell-Scrodinger equations. They are automatically

obtained by applying the time-dependent variational principle to the Lagrangians

(5.5)-(5.7). In the limit of a complete basis these equations are exact.

The complex phase space that carries the associated dynamics is endowed with
the Poisson bracket
T 1
SF/aO iC 0 0 0 0 0 o G/a*

aF/ay 0 -iC* 0 0 0 0 cG/ay
O{}F/Oa 0 0 0 0 -MT 0 OG/a
{F, G}
oF/a9 0 0 0 0 0 -NT G/a9

OF/9t7 0 0 M 0 0 0 OG/7

OF/0 0 0 0 N 0 0 OG/c
(5.17)









Even though the symplectic form is not canonical, its inversion is simple. The matrix
elements in w involve gaussian overlap integrals like (Gzj Gk) = J,(GZx)(xIGk)d3x.
5.2 Maxwell-Schr6dinger Theory in a Real Basis

As was done previously, each dynamical variable may be expanded into a com-
plete basis of functions GK as

Q(x, t) = EYcGkc(x) qc(t) P(x, t) = EYGc(x)pc(t)
Ak(x, t) EkcG(x) ak(t) fk (x, t) EkG(x)7kKl(t) (5.18)
S(x, t) = E,: Gk(x) O (t) e(x, t) -= E Gk (x) Ok(t),

where the index IC runs over the basis and the index k runs over 1, 2, 3 or x, y, z.
Unlike in (5.3), the coefficients in (5.18) that carry the dynamics are all real-valued.
In this basis, the real Lagrangian densities become

LMa~x = EZ K[(aO/amM)am (./07.mM)i-mM]SMax HMax (5.19)



LSch = Zc [(/qik)qK: (/9piK)pC]SSch HSch (5.20)

1
Lgauge = YICM [(/a 9k) c (-/a0k)0k]Sgauge Hgauge (5.21)

with integrals

SMax Jv Ad3X SSch f PQd3x Sgauge f= VfV d3x. (5.22)

Applying the calculus of variations to the above Lagrangians leads to the equa-
tions of motion:

02SMax H.
-an, = or MmM,maNA = VMm H (5.23)
0'mMooH,,\r 0rmMn


02Sgauge OH
0Ogauge O = or Nzc =- VozH (5.24)
80-ia -a7