Group Title: study of accidental degeneracy in Hamiltonian mechanics
Title: A Study of accidental degeneracy in Hamiltonian mechanics
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Title: A Study of accidental degeneracy in Hamiltonian mechanics
Physical Description: vi, 108 leaves : illus. ; 28 cm.
Language: English
Creator: Dulock, Victor August, 1939-
Publication Date: 1964
Copyright Date: 1964
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Subject: Dynamics   ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- UF
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non-fiction   ( marcgt )
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Thesis: Thesis--University of Florida, 1964.
Bibliography: Bibliography: leaves 105-107.
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General Note: Manuscript copy.
General Note: Vita.
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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: alephbibnum - 000568326
oclc - 13645229
notis - ACZ5051

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A STUDY OF ACCIDENTAL DEGENERACY

IN HAMILTONIAN MECHANICS













By
VICTOR AUGUST DULOCK, JR.


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












UNIVERSITY OF FLORIDA

June, 1964
























In memory of John F. Kennedy
















ACKNOWLEDGMENTS


The author wishes to express his sincere appreciation to his

supervisory committee and in particular to Professor H.V. Mclntosh,

who directed this research. He would also like to acknowledge the

counsel and guidance given him by Dr. B.S. Thomas throughout his

graduate program.

The author also wishes to thank the Department of Physics and

the Graduate School of the University of Florida for their financial

support during his graduate career.

Finally, the author gratefully acknowledges the help and en-

couragement given him by his wife, Maryann.















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . . . . . . . . ... .... . ii

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

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

Purpose . . . . . . . . . 1
Historical Sketch . . . . . . . . . 2

CHAPTER

1. MATHEMATICAL FOUNDATIONS . . . . . . . 7

Introduction . . . . . . . .. . . 7
Lie Algebra Defined by Poisson Brackets . . . 8

2. THE HARMONIC OSCILLATOR . . . . . . .... 20

Introduction .................. . 20
The Plane Isotropic Oscillator . . . . .. 20
The Hopf Mapping . . . . . ..... 24
Canonical Coordinates ............... 29
The Anisotropic Harmonic Oscillator in
Two Dimensions . . . . . . . . ... 31
Commensurability and Constants of the Motion .... 34
The Isotropic Oscillator in n Dimensions . . .. 36
The Hopf Mapping . . . . . . . ... 38
Other Canonical Coordinates . . . . .... 42
The Anisotropic Oscillator in n Dimensions .... 46

3. THE HARMONIC OSCILLATOR IN A UNIFORM MAGNETIC FIELD . 50

Introduction . . . . . . . . . . 50
Classical Zeeman Effect for the Harmonic
Oscillator ..... . . . . . ... . 52
Limiting Cases . . . . . . ..... 55
Canonical Coordinates .. . . . . . . 59
Rotating Coordinates . . . . . . ... 61
Discussion of the Constants of the Motion . . .. 70
Gauge Transformations in Uniform Magnetic Fields . 77












TABLE OF CONTENTS--Continued

Page

CHAPTER

4. THE KEPLER PROBLEM ............... ... .. .82

The Two Dimensional Case in Polar Coordinates . 82
The Kepler Problem In Three Dimensions . . . .. 87
The Two Dimensional Kepler Problem In Parabolic
Coordinates ......... ... ........ .. .94

5. SUMMARY .... ................... 98

LIST OF REFERENCES ..................... .. 105

BIOGRAPHY . .. . .. . .. . .. . . .. 108















LIST OF FIGURES


Figure Page

2.1 Gnomonic Projection . . . . . . . ... 26

2.2 Inverse Stereographic Projection . . . . .. 27

3.1 Particle orbit In a strong uniform magnetic field
and harmonic oscillator potential . . . . .. 64

3.2 Particle orbit in a weak uniform magnetic field
and harmonic oscillator potential . . . . .. 65

3.3 Particle orbit in a fixed uniform magnetic field
and harmonic oscillator potential with a large
initial tangential velocity . . . . . ... 66

3.4 Particle orbit in a fixed uniform magnetic field
and harmonic oscillator potential with a small
initial tangential velocity . . . . . ... 67

3.5 Particle orbit shown in Fig. 3.1 as viewed from a
rotating coordinate system . . . . . .... . 71

3.6 Particle orbit shown in Fig. 3.2 as viewed from a
rotating coordinate system . . . . . .... 72

4.1 Three dimensional coordinate system for the Kepler
problem . . . . . . . . ... .. ... .90















INTRODUCTION


Purpose


One ordinarily defines the accidental degeneracy of a system as

that which does not follow from an analysis of the obvious geometrical

symmetry of the system in configuration space. Frequently, neverthe-

less it can be shown that there is a higher order symmetry group in

the phase space of the system, formed with the help of certain hidden

symmetries, which is adequate to account for this accidental degen-

eracy. A well known example of a system possessing this property,

and one of the first to be investigated historically, both classically

and quantum mechanically, is the non-relativistic Kepler problem, whose

quantum mechanical analogue is the problem of a one-electron atom.

In the quantum mechanical case, the invariance of the Hamiltonian

with respect to the three dimensional rotation group accounts for the

degeneracy in the "m" quantum number, but not for that in the "1"

quantum number.

The purpose of this work is to review the more important results

obtained on the accidental degeneracy of harmonic oscillators and the

Kepler problem and to find some general principles with which the

symmetry groups of Hamiltonian systems can be found. In Chapter 1 a

basic mathematical foundation is laid. This foundation is based on

the fact that all quadratic functions over the phase space of a system

form a Lie algebra. On this basis a method is developed which enables

1












one to find, rather easily, the constants of the motion for systems

with quadratic Hamiltonians or for those which can be made quadratic

through a canonical transformation.

This method is then applied to various systems in the succeeding

two chapters. The harmonic oscillator is treated in Chapter 2, both

in the two and in the n dimensional case. Chapter 3 deals with a

general Hamiltonian which may be interpreted as that of a charged

mass point in a plane harmonic oscillator potential and uniform mag-

netic field. Finally, in Chapter 4 the Kepler problem is discussed

and found to have an SU3 symmetry besides the R4 generally associated

with it.

All calculations are carried out classically. The results ob-

tained seem to be valid in the quantum mechanical case except as noted.

The transition to the quantum mechanical version is carried out in the

usual manner by replacing Poisson bracket relations with commutator

brackets and replacing functions by their corresponding quantum mechani-

cal operators.




Historical Sketch


W. Pauli [1] was perhaps the first to associate the accidental de-

generacy of the Kepler problem with the less well known second vector

constant of the motion, the Runge vector R, which had been discussed

earlier by W. Lenz [2]. Pauli defines this vector classically as lying

in the plane of the orbit with a direction from the force center to the











aphelion and a length equal to the eccentricity of the classical orbit

and he also gives the quantum mechanical operator definition. The

commutation rules between the components of this vector and the angular

momentum are also stated explicitly. As a consequence of the the com-

mutation rules he shows that the system is degenerate in "1" also.

However, L. Hulthen [3] observed the fact that these commutation rules

are the same as those of the generators of the four dimensional rotation

group. 0. Klein was given credit for first recognizing this fact.

V. Fock [4], in 1935, using a stereographic projection in momentum

space, solved the Schridinger equation in momentum space and showed

that the solutions are spherical functions of the four dimensional

sphere and hence that the symmetry group is R4. Later V. Bargmann [5],

in a discussion of Fock's results, showed that Fock's group was gener-

ated by the components of the two vector constants R and L, the angular

momentum.

It was about this time that 0. Laporte [6] obtained a differential

operator identical to the Runge vector and presented a method for ob-

taining all the eigenfunctions of the hydrogen atom by differentiation,

similar to the method used to find the eigenfunctions of L2 by using

the raising and lowering operators L+ and L_. He made extensive use

of the stereographic parameters, developed by him and G.Y. Rainich [7]

about the same time, in momentum space. The wave functions were found

to be proportional to hyperspherical surface harmonics.

In 1939 and 1940 E.L. Hill and his student, J.M. Jauch [8, 9, 10]

published two articles and a Ph.D. thesis dealing with accidental de-

generacy. These authors considered both the two and three dimensional











Kepler problem and the harmonic oscillator in two and three dimensions.

The Kepler problem was found to have R3 and R4 symmetry groups in two

and three dimensions respectively, while the harmonic oscillators had

the symmetry of SU2 and SU3, the groups being generated by the constants

of the motion in all cases. The transformations induced by the gener-

ators were found to have an analogy in classical mechanics, namely that

they represented transformations of one orbit of phase space into

another of the same energy. They also showed the existence of a corre-

spondence between transformations in classical mechanics and quantum

mechanics.

In 1949, A.W. Saenz [11] wrote a Ph.D. thesis under Laporte, in

which the symmetry groups explaining the accidental degeneracy of vari-

ous problems were found by reducing the problems to force free motion

on hyperspherical surfaces of various dimensions. In this manner he

considered the Kepler problem, harmonic oscillator and rigid rotor.

More recently, the n dimensional isotropic oscillator has been

discussed by G.A. Baker [12]. He found the synmetry group to be SUn

and demonstrated that this group could account for all the degeneracy

found. This had been demonstrated earlier by Y.N. Demkov [13] from a

slightly different point of view. Whereas Baker attacked the problem

by finding the most general group which left the Hamiltonian invariant,

Demkov used the generators of infinitesimal transformations which com-

muted with the Hamiltonian and then showed that this defined an SUn

group. In a later article Demkov [14] demonstrated the equivalence of

his and Baker's works.











The connection between accidental degeneracy and hidden symmetry

was confirmed by A.P. Alliluev [15] in 1958. He considered two ex-

amples, the two dimensional oscillator and the Kepler problem in n

dimensions, which reduced to Fock's result in the case where n = 3.

In the general case, the hidden symmetry of the Kepler problem in n

dimensions was found to be that of the n + 1 dimensional hypersphere.

The one dimensional Kepler problem has been treated by R. Loudon [16]

and was found to be doubly degenerate in all levels except the ground

state, an apparent violation of the theorem that the energy levels of

one dimensional systems are nondegenerate. Loudon discusses his results

in the light of this theorem and shows that it holds only when the po-

tential is free of singularities. In 1959, H.V. Mclntosh [17] wrote a

qualitative review of the work up to that time and stated the results.

The most recent article to appear is one by Demkov [18] in which the

concept of excessive symmetry groups is introduced. He gives a pre-

scription for finding the minimal symmetry group and as an example

applies it to the anisotropic two dimensional oscillator with incom-

mensurable frequencies.

In addition to the works cited above dealing with thestudy of

accidental degeneracy itself, there has been a considerable amount of

work in using the symmetry groups for various problems and applying

them to the many body problem, particularly to the shell model of

the nucleus. One of the first of these was by J.P. Elliott [19, 20]

who undertook to study collective motion in the shell model utilizing

a coupling scheme associated with the degeneracy of the three dimensional










harmonic oscillator. The direct result was a classification of the

states according to the group SU3. Following this treatment was a

series of papers by V. Bargmann and M. Moshinsky [21, 22] on the

group theory of harmonic oscillators, in which they developed a

classification scheme for states of n particles in a common harmonic

oscillator potential. They were primarily concerned with finding a

scheme which would exhibit explicitly the collective nature of the

states. Similar work was also done by S. Goshen and H.J. Lipkin

[23, 24] with the exception that the particles were in a one dimen-

sional potential well. This system could also be described from a

collective viewpoint and these states could be classified using the

group of transformations.

In recent times, L.C. Biedenharn has extensively studied the

Kepler problem and its symmetries, both in the non-relativistic and

relativistic cases. In the non-relativistic [251 case he has obtained

relations for the representations of R4 exploiting the fact that R4 is

locally isomorphic to R3 x R3. The non-relativistic problem has also

been considered by Moshinsky [26], where use has been made of the

accidental degeneracy in the coulomb problem to obtain correlated wave

functions for a system of n particles in this potential. In the rela-

tivistic case, Biedenharn [27] and Biedenharn and Swamy [28] have ob-

tained operators which are analogous to the angular momentum and Runge

vector in the non-relativistic problem.













CHAPTER 1


MATHEMATICAL FOUNDATIONS


Introduction


In classical and quantum mechanics "quadratic" Hamiltonians play

an exceptional role, a fact which is traceable in the end to the fact

that the Poisson bracket operation satisfies axioms whose expression

is tantamount to saying that the Poisson bracket is an alternating bi-

linear functional which is a derivative in each of its arguments.

If H(n) designates the vector space of homogeneous polynomials

of degree n, whose basis may be chosen to be composed of homogeneous

monomials, the Poisson bracket operation maps the cartesian product

H(m) x H(n) into H(m+n-2), as will be shown later. The Poisson bracket

operation is a bilinear functional, which means that it defines a linear

transformation when either argument is held fixed, and so it always has

a matrix representation. For the case m = 2, one has a mapping from

H(n) into itself, representable by a square matrix. In this latter case,

the extensive lore of matrix theory is available to help discuss the

transformations. In particular, the mapping may be described in terms

of its eigenvalues and eigenvectors.

Thus the mapping

Tq(p) = (q, p), (1.1)


for a fixed q E H(2), and argument p H(n), can be expected to have

7











specialized properties which become all the more significant when

n = 2. In this case the elements of H(2) form the basis of a Lie

algebra under the Poisson bracket operation. In any event, the gen-

eral mapping Tq of Eq. (1.1) yields representations of this algebra

as it acts on polynomials, homogeneous of various degrees as shall

be shown later for the case n = 2. Thus the allusion to the excep-

tional role of quadratic Hamiltonians refers to the expectation that

the algebraic structure of the Lie algebra will have its reflection

in the dynamical properties of the system which it represents.




Lie Algebra Defined by Poisson Brackets


The set of 2n linearly independent coordinates and moment of an

arbitrary physical system is said to form its phase space (. These

variables may be paired in such a way that they can be indexed by fi,

i = n, ...1, -1, ...-n, where the positive indices indicate coordinates

and the negative ones the corresponding conjugate moment. The Poisson

bracket of any two functions of these variables is defined by


n
(g,h) = 6g 9 (1.2)
1 -1 -I I



It follows from Eq. (1.2) that the Poisson bracket operation satisfies

the axioms of a Lie algebra [29, 37]:


(f,g) = (g,f) (alternating rule),


(1.3)











(f, ag + Bh) = a{f, g) + B(f, h) (linear), (1.4)


(f, (g, h)) + (g, (h, f)) + (h, (f, g)) = 0 (Jacobi identity).

(1.5)

The Poisson bracket also acts like a derivative in that it satisfies

the relation

if, gh) = g(f, h) + (f, g)h (1.6)


Although all second degree polynomials form a Lie algebra, the

linear monomial functions of the coordinates in phase space itself

also form a linear vector space 0, where the basis may be chosen to

be the 2n fi. Denoting the Poisson bracket operation ( ) as the

symplectic inner product in 4, it is noted that this product does

not satisfy the usual axioms for an inner product since it is anti-

symmetric under interchange of the arguments. This difficulty may be

remedied in the following manner. Using the definition of the Poisson

bracket it is found that


1 i = -j >0

(fi f = 0 i f -j

-1 i = -j < 0 (1.7)


From this it is observed that if i > 0, the dual vector fi to f may

be taken to be f-1. Similarly for i < 0 the dual vector is taken to

be -f .. Thus the dual basis to
-I


(1.8a)


T = (fn, ... fli f-i .... f-n)










is then
r = (-f- ... -f-1 .fl' fn). (1.8b)


With these definitions the inner product may be defined as


(f, g) = (f, g), (1.9)


where f and g are linear combinations of the fi and


f= = afi (1.10)
i


It is evident then that

(fi' fj) = (fi' fj) = ij (1.11)

and

(f, f) = f, f) = ai2 (1.12)



The space D also has the peculiar property of being self-dual in a

special way, namely the mapping Tf is a linear mapping and is in a one

to one correspondence to the vectors in the dual space D+.

Considering tensor products of @, it is observed that (f, g) also

defines a mapping on these products. If one defines

H(n) = 0 (D (&) .... ( = (e


and if f E H(m) and g E H(n), then (f, g) E H(m+n-2). In particular,

when m = 2 i.e., f E Q = H(2), a mapping is obtained which is represent-

able by a square matrix.










Such a mapping may be expected to have eigenfunctions and eigen-

values. Let h be a fixed element of Q (a quadratic Hamiltonian) and

let gi and Xi be the eigenfunctions and eigenvalues such that


(h, gi) = Th(gi) = Xigi, (1.14)

where the gi are taken to be linear i.e. gi E H(1). If h is represent-

able as a normal matrix, the gi form a basis for the space H(1), and

are simply linear combinations of the fi. In particular, since the

gi E D = H(1), they form a basis for all the tensor spaces as well
where all possible products of the gi are taken i.e.,


((gi)) = H(1)

((gig ))= H(2)



((g .... gm)) = H(n). (1.15)

On account of the derivative rule Eq. (1.6) the following relation is

satisfied

(h, gij = gi(h, gj) + (h, gi) gj

= (Xi + Xj) gigj, (1.16)

and hence the monomials in gi are eigenfunctions of h in every H(n)

and form a basis even in the case of degeneracy, for all possible eigen-

functions.

In cases where the operator Tf is normal, a complete set of or-

thogonal eigenvectors exists which will be denoted gi. Hence, if the











eigenfunctions are indexed such that gi = g-i then the eigenvectors of

any normal operator satisfy the rule


(gi, 9j) = i,-j. (1.17)


The eigenvalues Xi corresponding to the eigenvectors gi of the normal

operator h occur in negative pairs. This follows from a consideration

of the Jacobi identity


(h,(g ,i gi )) + (g1,' (gi h)) + (gi,(h, gi} = 0,

thus
(h, 1) + (-Xi) (g ,i' gi) + i (gi, g_- ) = 0,

and
(X-i + Xi) fgi, g-i) = 0, (1.18)


and hence, i. = --i since (g., g-i) i 0.

Since h H(2), h may be written as

n
h = C ijgigj, (1.19)
i,j=-n
i j

where the gi are eigenfunctions of h belonging to eigenvalues Xi and

are indexed so that
Xi = -X i (1.20)


Assuming that the eigenvalues of h are distinct and using the identity

(h, h) = 0 one has, with the aid of Eq. (1.19)











0Z
i,j
i=j




-z
i,j
i'j


Cij h, gigj)


Cij (xi + .j) gig..
Ii I J J'3 3


(1.21)


Hence each coefficient must vanish and either Cij = 0 or (Xi + Xj) = 0.

However Xi = -Xj only if j = -i. Therefore


(1.22)


To further evaluate these coefficients recall that


(h, gk) = .kgk,


thus


(1.23)


n

k'k = Ci-i gig-i' 9gk
i=l


n
= ) C [g (9g 9, + 9-i i k]9 9 "
i,-ii E. '-i 9k + g- g k
i=l

Using the fact that the coefficients C.. are symmetric in the

the coefficient of gk on the right side is


xk = Ck,-k g-k' gk)


(1.24)



indices


(1.25)


h =X Ci,-i gig-i.












and hence
C (1.26)
cI-I = g-i gi)


The final conclusion is that



h = g i9-i- (1.27)
-(g g;)
i=l


Consider now the case where h may have degenerate eigenvalues.

Using the general form for h, Eq. (1.19), and calculating (h, h) as

before Cij = 0 unless Xi = -JX. However in the present instance it is

possible to have Xk = 'm, k # m and hence Xk m= -'.- Hence

n n
h = C ii gigi Ck,m gkg-m, (1.28)

i=l k,m=l
=x
k m
k m


Calculating once again (h, gj) results in the following equations:

n n
Xj9j C g Ck,-m gk C -m gj} (1.29)
i=l k,m=i
Xk=Xm
k m
k m

n

= i -i [gi g-i' gj} + g-i gi' gj)]
i=l










n

+ Ck,-m [gk9-m' j)+ -mk' gj ]
k,m=1
Xk=
k m


Consider the first term in the second sum. If nmj then kfm implies

k#j and hence Ck,- must be zero. Therefore, as before

n
h = -'- gig-i (1.30)
(9-i, 9i)
i=l

It should also be noted that the transformation to eigenvector

coordinates for h is a canonical transformation since the eigenvectors

satisfy (gi, g) = 5i,-j, which is a necessary and sufficient condition

for g. and gi to be canonical variables.

Consider now the canonical form of a non-normal quadratic operator

h, of the Jordan canonical form. Again

n n
h = C Cjgj, (1.31)

i=-n j=-n

i>_j


where the gi are the principal vectors of h and indexed such that the

same orthogonality conditions hold as before. Using the property that


(h, gi} = gi+1, (1.32)


it is found that










n n

gk+l = Ci
i=-n j=-l
i>-j



z C ij
i=-n j=-n
ij


Ck+l,-k Sk+l


C-k,k+l gk+l


9{gigj 90k




[gigj gki + gj gi' i k





k > 0


k < 0 .


Hence all Cij = 8i,j+_ and

n
h= gg-i+l 1
i=l


The sympletic norm also produces certain symmetries and anti-

symmetries. For example, the dual mapping R(fi), which maps the

coordinates into moment and the moment into the negatives of the

coordinates,

R(fi) = f-


i > 0


R(f i) = -fi


preserves the inner product; i.e.


(R(fi). R(fj)) = (fi fj)


On the other hand the mapping P(fi) which simply exchanges coordinates

and moment


(1.33)


(1.34)


(1.35)


(1.36)












P(fi) = f-i, (1.37)

or the time reversal mapping T(fi)


T(fi) fi

i > 0 (1.38)
T(f-i) = -f-i


reverses the sign of the symplectic norm


(P(fi)' P(fj)) = (f, fj)


(T(fi), T(fj)) = (fi fj} (1.39)


These operators may be expressed in terms of the eigenvectors, gi of

the normal operator h, previously defined. Recall also that the gi

form a set of canonical coordinates and moment. Using the convention

that positive indices represent coordinates and negative indices the

corresponding canonical moment, these operators have the following

canonical forms:
n
R = ggi +g-i9-i (1.40)
2(9_ gi


n

i=1
P gig; g ;g ;
=1 2(gi, g-i)


n
T = gg-' (1.42)
i=1 ^-i' gi











These operators have the following effects on

gkTT(
TT 9k) = (T, gk) =
-gk

Tp(gk) = Pgk) = g-k



TR(g = (R, gk = -k
g-k


The composite

equivalent to


eigenvectors of h:

k > 0
(1.43)
k < 0,


all k,


k > 0

k < 0


(1.44)


(1.45)


of any two of the operations on an eigenfunction is

operating with the third operator:


T p(TT (g9)) k= = TR(gk)
TR(TTk g-k k
TR(TT(gk)) = g-k = Tp(gk)


k > 0 TT
k < O = T(g).
k

It is also interesting to note that

following Poisson bracket relations


(P, T)

(R, T)

(P, R)


the three operators satisfy the

among themselves:


= 2R

= 2P

= 2T


(1.49)

(1.50)

(1.51)


The composite of the normal operator h with each of these operators

on the eigenvectors is also worth noting. It is relatively easy to show


gk

_gk


(1.46)



(1.47)


(1.48)


Tp(TR(gk)) =











that
Th(TT(k)) = TT(Th(gk)), (1.52)

and hence the matrices representing h and T commute. However one finds

that P and R anticommute with h, i.e.


T (Th(gk)) = Th(T (gk)) (1.53)

and
T (T (g)) Th (T (g )). (1.54)
Rhk hRk


The existence of either one of these last equations is sufficient to

insure that the eigenvalues appear in negative pairs [30].

In the next two chapters the method described above will be applied

to harmonic oscillator Hamiltonians in two and n dimensions and to vari-

ations on the oscillator Hamiltonian.















CHAPTER 2


THE HARMONIC OSCILLATOR


Introduction


The accidental degeneracy and symmetries of the harmonic oscillators

has been discussed by several authors. A particularly thorough account

has been given in the thesis by J.M. Jauch [9], in which he considers the

isotropic oscillator in two, three and n dimensions, and finds the sym-

metry group to be SUn, i.e., the Hamiltonian is invariant under transfor-

mations induced by the generators of the group. A short discussion of

the anisotropic oscillator in two dimensions with commensurable frequen-

cies is also given. The main results of this work are also contained in

the notes of E.L. Hill [33]. More recent discussions of the n dimensional

isotropic oscillator have been given by G.A. Baker [12] and Y.N. Demkov

[13, 14, 18].




The Plane Isotropic Oscillator


In units such that the mass and spring constant are unity, the

Hamiltonian of the plane isotropic harmonic oscillator is


H = (P2 + 2 + + y2)/2 (2.1)
x y

With respect to the basis (x,YPx ,Py), the matrix representation of H











as an operator under the Poisson bracket relation is,


0 0
S= 0 0
-1 0
0 -1


0
0
0


(2.2)


This representation is found by calculating the effect of H on each

member of the basis under Poisson bracket. With the convention that

x is a column matrix with a 1 in the first row, y with a 1 in the

second row and so on, the matrix representation of H is then easily

constructed. The eigenvalues of the matrix H are


x = i


(2.3)


each root appearing twice.

with X = + i are


a = (1/I2)(Px ix) = 1/J2


The normalized eigenvectors associated


\-i


b = (l/f2)(Py iy) = 1/F2


0



(2.4)
(2.4)


and those associated with X = i are


a* = (I/F2)(Px + ix) = 1/W2


0o b* = (l/F2)(Py + iy) = 12 i



(2.5)


where the column matrices are the matrix representations of the

functions on the left. One may easily verify this representation,

for example












(H, a) = (H, Px ix) = i(Px ix) (2.6)


while multiplying the matrix in Eq. (2.2) into the column matrix repre-

senting a also gives i times the same column matrix. The quantum me-

chanical analogues of these four quantities are the well known raising

and lowering operators for the harmonic oscillator. With these four

quantities ten linearly independent quadratic monomials can be con-

structed which are eigenfunctions of H and have as eigenvalues 0, + 2i

or 2i. Only four of these are of interest here, namely those with

eigenvalue zero. They are the four linearly independent constants of

the motion (since they commute with H under Poisson bracket) listed

below:

aa* = (P2 + x2)/2 (2.7)
x

bb = (P2 + y2)/2 (2.8)
y

ab" = [(PxPy + xy) + i(yPx xPy) /2 (2.9)


a*b = [(PxPy + xy) i(yPx xPy)]/2 (2.10)


For the purpose of physical interpretation it is more convenient to

deal with the real and imaginary parts of these constants separately,

as well as to separate the Hamiltonian from them. Accordingly the

following four quantities are introduced:


H = aa* + bb* = (P2 + P2 +2 + y2)/2 (2.11)
x y


D = aa bb = (P2 + x2 P2 y2)/2
x y


(2.12)












L = i(a*b ab*) = yPx xPy (2.13)


K = a*b + ab* = xy + PxPy (2.14)


These four constants have the following simple interpretations.

a) H, the Hamiltonian, is the total energy of the system.

b) D is the energy difference between the two coordinates.

c) L is the angular momentum of the system and the generator of

rotations in the xy plane.

d) K is known as the correlation and is a peculiar feature of the

harmonic oscillator. As a generator of an infinitesimal con-

tact transformation, it generates an infinitesimal change in

the eccentricity of the orbital ellipse while preserving the

orientation of the semlaxes, and preserving the sum of the

squares of their lengths. The energy of a harmonic oscillator

depends only on the sum of the squares of the semiaxes of its

orbital ellipse, which remains constant under such a transfor-

mation.

The set of functions (K, L, D) is closed under the Poisson bracket

operation. Explicitly, their Poisson bracket table is


K L D

K 0 2D -2L (2.15)

L -2D 0 2K

D 2L -2K 0


Aside from the factor 2 these are the Poisson bracket relations of the












generators of the three dimensional rotation group, or the three com-

ponents of the angular momentum in three dimensions. Since the con-

figuration space of the two dimensional isotropic oscillator is appar-

ently only two dimensional and has only rotations in the xy plane as

an obvious symmetry, the occurrence of the three dimensional rotation

group is rather anomalous.

By performing a series of geometrical transformations on a, a b

and b* one can explicitly demonstrate the spherical symmetry of the

system. This series of transformations is sometimes called the Hopf

mapping.




The Hopf Mapping [31]


Perhaps the best analytic representation of the Hopf mapping is

obtained by introducing polar coordinates (a, r, o, p) for the complex

variables a and b. Explicitly, these are


a = (Xe-ip cos T)A/-2 (2.16)


b = (\e-io sin 1)AT2, (2.17)


where X does not denote the eigenvalue of H as previously. The first

step in the Hopf mapping is to form the ratio w defined as the quotient


S= X2 a/b (2.18)


The formation of this ratio may be regarded as a gnomonic projection of

the four dimensional space, regarded as having two complex dimensions,












onto a two dimensional space (Fig. 2.1), having one complex dimension.

This maps the point (a,b) lying on the two dimensional complex circle

of radius X2 into the point


\2 a/b = \2 ei(O-P) cot r (2.19)


on the complex line.

The second step is to regard the complex point w = X2 ei(0a-) cot

as not lying on a complex line; but rather as a point in a real two

dimensional plane. One now performs an inverse stereographic projection

onto the surface of a three dimensional sphere whose south pole is tan-

gent to the plane at its origin. By choosing the origin of the three

dimensional space to be at the center of the sphere, a point on the

surface of the sphere will be specified by giving the radius r, the

azimuth 4 and the colatitude 0. For convenience the sphere is chosen

to have diameter X2. The azimuth is measured in a plane parallel to the

original plane and hence

0 = o p (2.20)


Thus to determine the location of the projected point P on the sphere,

only the colatitude 0 is left to be determined. From Fig. 2.2 one has


tan (A 2) = cot T (2.21)
2 2
but

tan ( = cot (2.22)

hence


0 = 2T .


(2.23)







26










N



3


















0








27







0
u




















c/C
"4

a






















L.
O.
U

t-
O



0




4/
(O
C)
1I
LC
;)

I (-



c&

o












Thus the coordinates of the point P projected onto the surface of

the sphere from the point w = X2 ei(o-p) cot T are


0 = 2Tr


S= O p


r = \2/2 .


(2.24)


(2.25)


(2.26)


Expressing the quantities H,

these new coordinates gives


K, L and D in Eq. (2.4) in terms of


H = X2/2


K = (X2/2) sin 0 cos 0 = X


L = (X2/2) sin 0 sin P = Y


D = (X2/2) cos 0 = Z .


Hence, K, L and D simply determine a point on the surface of this sphere.

It should be noted that K generates infinitesimal rotations about X, L

about Y, and D about Z.

Since these three constants of the motion are then directly the

coordinates of the fixed point corresponding to the orbit of the oscil-

lator, it is apparent that they generate rotations of the sphere, which

transform one orbit into another of the same energy.


(2.27)


(2.28)


(2.29)


(2.30)












Canonical Coordinates


Whenever two functions satisfy the Poisson bracket relation


(A, B) = LB (2.31)


where 4 is a scalar, a corresponding equation,


(A, (1i ) In B) = 1 (2.32)


may always be written, which is the rule satisfied by a coordinate and

momentum which form a canonically conjugate pair. The first of these

two equations simply states that B is an eigenfunction of A with respect

to the Poisson bracket operation. Consequently it is to be expected

that such eigenfunctions be a ready source of canonical variables. A

system in which the Hamiltonian is one of the moment can be obtained

in the present case. Consider the following diagram in which the two-

headed arrows imply that the Poisson bracket of the two functions is

zero and the one-headed arrows imply that the Poisson bracket is + 1

in the direction indicated:


H = a a* + bb* ( D = a a* bb




(1/4i) In (a*b*/ab) <-- (1/4i) In (ab*/a*b).

(2.33)

If these four quantities are rewritten in terms of the four parameters

introduced in the Hopf mapping and p + o is denoted by T, then Eq. (2.33)

becomes











H = X2/2 > D = (X2/2) cos 0


SxI
T/2 < 0/2 (2.34)


Hence the Hamiltonian and the energy difference are the two moment

and the angles F/2 and 0/2 are the corresponding canonical coordinates.

Since '/2 is conjugate to H, T increases linearly with time.

The final result of this transformation is to map an entire orbit

into a single point. Since the radius of the image sphere determines

the energy, the only information lost as a consequence of the mapping is

the phase of the point representing the harmonic oscillator in the great

circle comprising its orbit. The lost phase may be recovered by attach-
ing a flag to the point representing the orbit, which will then rotate

with constant velocity. The missing angle may then be taken as the

angle which the flag makes with its local meridian. In this way the

motion of the harmonic oscillator in its phase space is shown to be

equivalent to the motion of a rigid spherical rotator, and in fact the
parameters 0, 0 and Y are just the Euler angles describing the orien-

tation of the rotor. These same parameters may also be used for the

description of spinors, and a particularly lucid geometrical interpre-

tation of this transformation may be found in a paper of Payne [32]

describing various methods of representation of two-component spinors.












The Anisotropic Harmonic
Oscillator in Two Dimensions


The anisotropic oscillator, both with commensurable frequencies

and with non-commensurable frequencies has been discussed by Jauch and

Hill [10] and also by Hill [331. These authors were able to show that,

In the case of commensurable frequencies, the symmetry group of the

system is again the three dimensional rotation group. However, they

were unable to come to any conclusion concerning the case with incom-

mensurable frequencies. Recently R.L. HudsonI has been investigating

the incommensurable case also.

It can be shown though, that even with non-commensurable frequen-

cies the symmetry group is also a three dimensional rotation group.

The proof is sketched below and follows closely that used in the iso-

tropic case, except where noted.

If units are chosen such that the frequency in the x coordinate is

unity, the Hamiltonian is


H = (P2+ P2 + x ++ 2y2)/2 (2.35)
x y

where w is the frequency in the y coordinate and is an irrational number.

The eigenfunctions and eigenvalues of H under the Poisson bracket opera-

tion are listed below:


IPrivate communication.











Eigenfunction Eigenvalue

a = (l1/2)(Px ix) i

a* = (1I/2)(Px + ix) -i

b = (l/A2)(Py iwy) iw

b*= (lJ2)(Py + iwy) -i (2.36)


Although the most obvious four constants of the motion are aa*, bb ,

(a) b*, and (a*)Wb, a more convenient set is the following collection of

independent combinations of them:


H = aa* + bb* (2.37)


D = aa* bb* (2.38)


K = [(a)Wb* + (a*) b)/(aa*)(w-1)/2 (2.39)


L = i[(a)b* (a*)wb]/(aa*)(w-)/2 (2.40)


Again the set (K, L, D) is closed under the Poisson bracket operation

and forms a Lie group, the three dimensional rotation group. The

Poisson bracket table is listed below:
K L D

K 0 2wD -2wL

L -2wD 0 2wK

D 2wL -2wK 0 .(2.41)


A mapping to polar coordinates as in the isotropic case can once

again be performed, however it is not a Hopf mapping. The form of the

final equations for the constants is the same as in Eqs. (2.27) to










(2.30), however, for the anisotropic case

0 = up o (2.42)

A set of functions which proves to be convenient to use as the new
canonical coordinates is diagrammed below with the same convention as
before:
H D



I I
I I(a) wb%*\ 1 (aI*) (a)w
4it (a)(b 4iw (a)b* (2.43)

If the parameters from the mapping are substituted into Eq. (2.43)
this table becomes identical to Eq. (2.34) with the conditions that 0
is defined by Eq. (2.42) and

T = wp + o (2.44)

It is rather interesting to note that the entire effect of the
possible incommensurability of the frequencies in the two coordinates
is completely absorbed by the power mapping (a)W, which is well defined
as exp (w In a) in the case of an irrational w. In this latter case
one has a mapping with infinitely many branches rather than a finite
number as in the case of a rational w. It is of course on this account
that the orbit--a Lissajous figure in the x-y plane--would then be open,

and would not correspond to periodic motion.












Commensurability and Constants of the Motion


In reviewing the analysis of the theory of the plane harmonic

oscillator, it is probably appropriate to draw attention to a number

of statements made, both in the literature of classical mechanics,

and the lore of accidental degeneracy. First of all, the existence of

accidental degeneracy is often considered to be allied with the exist-

ence of bounded, closed orbits. When this argument is used, it is

applied to the existence of algebraic integrals of the motion, and in-

deed it should be noted here that as long as the frequencies of the two

coordinates are commensurable, bounded closed orbits result and the in-

tegrals are algebraic, as required by the theory. However, the above

results also verify something which has been generally known [34], namely

that non-algebraic integrals can exist, and as would be expected, are

associated with space-filling motion.

Another point regards the separability of the Hamilton-Jacobi

equation in several coordinate systems. It is pointed out that if a

Hamilton-Jacobi equation is separable in more than one coordinate system,

it is necessarily degenerate [351. The argument usually given concerns

the fact that orbits with incommensurable frequencies in one coordinate

system will be space-filling curves, the boundaries of which are co-

ordinate arcs. Thus, in whatever coordinate system the motion is de-

scribed, the system of separation is uniquely defined as that one form-

ing an envelope of the various orbits. Of course, the option left in

the enunciation of this theory is the fact that changes of scale may

still be made, in which the new coordinates are functions each of only











one of the old coordinates, and not several or all. So it is in the

polar mapping, where one forms a ratio of powers of functions of only

a single coordinate. However, this change of "scale" seems to eliminate

the problem of the incommensurable frequencies at the same time, and

thus circumvents the "proof'of uniqueness. This result has only been

secured at a price, namely the inverse of the transformation is infin-

itely multiple valued [36]. Otherwise, one could easily reduce the

anisotropic oscillator to a static point on the Riemann sphere, return

from this to an isotropic oscillator, and separate in whatever system

desired.

These considerations are not unrelated to the problem of finding

a Lie group generated by the constants of the motion. In this respect,

a Lie group has always been regarded as being more important quantum

mechanically than classically, because its irreducible representations

would be associated with accidental degeneracies of the Schroedinger

equation. Since the anisotropic harmonic oscillator has no accidental

degeneracies in the case of incommensurable frequencies, it is some-

what surprising to find that the unitary unimodular group is still its

symmetry group, at least classically.

In the case of commensurable frequencies, proportional say to m

and n, the operator amb*n commutes with the Hamiltonian, and is inter-

preted as creating m quanta in the first coordinate and annihilating n

quanta in the second coordinate. Thus, m quanta in one coordinate are

equal to n quanta in the second coordinate, the energy is restored to

its original level, and the system is left In another eigenstate.












Thus, it is possible to expect and explain degeneracy in the case of

commensurable frequencies.

In the classical incommensurable case, the symmetry group still

exists. After all, the total energy of the oscillator is the sum of

the energies in each coordinate, which is reflected in the fact that

it is the sum of the squares of the semiaxes of the orbit (A Lissajous

figure, like an ellipsoid, has principal axes), and not the individual

semiaxes, which determine the energy. Thus, the dimensions of the

bounding rectangle of a space-filling Lissajous figure may be changed

subject to this constraint. This, in addition to an adjustment of the

relative phases in the two coordinates, is the effect of the constants

of the motion acting as the generators of infinitesimal canonical trans-

formations.

The problem of carrying this result over to quantum mechanics seems

to be in finding a proper analogue of the power functions (a)' and b,

which as ladder operators may not possess fractional powers, even though

they did so when considered as functions of a complex variable.




The Isotropic Oscillator in n Dimensions


Using units such that both the mass and force constant are unity,

the Hamiltonian for this system is



H = (P? + x?)/2. (2.45)
i=1












The eigenfunctions of H under the Poisson bracket operation are


aj = (Pj ixj)/-2 (2.46)

with eigenvalues

X = i (2.47)

as well as

a= (Pj + ix.)/T2 (2.48)

with eigenvalues

\ = i (2.49)


Hence one can immediately construct n2 linearly independent constants

of the motion of the form aia', i = 1, ...n and j = 1, ...n, although

as before, it is somewhat more convenient to deal with their real and

imaginary parts separately; in particular,

n
H = aia (2.50)
i=1


Di = ala ai;a i = 2, ...n (2.51)


K.. = a.a + a.a. j > i (2.52)
Ij i J I J

Lj = i[a'aj aiaj] j > i .(2.53)


The latter three sets of constants are closed under Poisson bracket

and satisfy the following commutation relations:











(Kj, Kk) = i[-(aam aiam)6k + (aa akaj)5im
-(aak aia)m + (a*a ama*) (254)
k jm m j m j ik (2.54)

(Li L il[(aa a al)jk + (aak a.a -)
I ma jk + k a)k im
(aak aiak)bjm (ajam aja)bik] (2.55)


(Di, Dj) = 0 (2.56)


(Kij Lkm) = (aam + a.aj k (ak + aia*)jm
+ (aa + a a*) (a a + a a*)bm (2.57)
1m i m jk j k j k im

(Lij, Dk) = (aiak + aj) k (a ak + aia ) k
(a a + aja)5il (2.58)


(D, Kkm) = i[(aak a a)bJm + (aa aa)J
(aa ala ) lk] (2.59)


Depending upon the values of i, j, k, and m all of the relations will

give linear combinations of the functions Di, Kij, or Lij as a result.

The relations also identify the symmetry group of the problem [37],

namely SUn where the n2 1 constants are considered as the generators

of the group.



The Hopf Mapping


In general only 2n-1 independent constants of the motion including
2
the Hamiltonian are expected to be found and not n2. Hence there should

exist n2-(2n-1) = (n-1)2 functional relationships among the n2 constants.












As a consequence, the following set of 2n-l constants have been chosen

for the Hopf mapping:

the (n-1) constants Klj where j = 2, ...n,

the (n-1) constants Llj where j = 2, ...n,


and the following linear combination of H and the Di

n
D = BIH + z ;Di (2.60)
i=2

where
2-n
1 2- (2.61)

and

B 2 (i = 2, ...n) (2.62)
n
In terms of the ai and ai
n
D = ala aia (2.63)
i=2

The (n-1)2 relationships between the constants are


LijKli LIIKIJ
Lij = H- +(2.64)



LliLl + KllKl(2.65)
J = H + D



(H+D)2 K 2 L 2
Di = li (2.66)
2(H + D)











n n
where n n
where H2 = D2 + KI + Li (2.67)

i=2 i=2


as may be verified by substitution from the definitions given in Eqs.

(2.50) through (2.53).

For the Hopf mapping let the 2n variables ai and af be redefined as


a1 =i 2(cos T)e"i (2.68)



a = e.(sin T)e-ij (j = 2, ...n) (2.69)
,J -2 J

where the Ej are direction cosines such that

n
E = 1. (2.70)
j=2

Forming the ratios
-- = Ej(cot -)e- ) (2.71)
aI


gives a complex projective space of n-I complex dimensions. Regarding

the space as having 2n-2 real dimensions and performing an inverse

stereographic projection a real space of dimension 2n-1 is obtained.

As in the case of the two dimensional oscillator the sphere of

2n-1 dimensions is considered tangent to the 2n-2 dimensional space

at its south pole. By defining 0 as the angle between a radius vector

from the center of the sphere and the diameter between the north and

south poles 2T can be identified with 0. The angles in the planes





41





perpendicular to the north-south diameter projected from the 2n-2 space

parallel to this line are identical with the angles in the space and

are denoted
Pi = 7i p (2.72)


The substitution of Eqs. (2.68) and (2.69) in Eq. (2.63) and into

the equations defining H, Klj, and Lij gives


H = (1/2)%2 (2.73)


D = (1/2)%2 cos 0 = Z (2.74)


Klj = (1/2)X2 sin OEj cos Vj = Xj (2.75)


Llj = (1/2)%2 sin Ocj sin qj = Yj (2.76)


where the Kli, Lli, and D have been identified with the 2n-l coordi-

nates in this space. As in the case for n = 2 these 2n-1 constants

determine a point on the surface of a sphere of radius H in this space.

At the same time the remaining constants still define infinitesimal

transformations in the respective coordinate planes such as rotations

or changes in the correlation or relative phases. It should be noted

that only in three dimensions is it possible to identify planes and

their perpendicular coordinates in a one to one fashion. In general

for n dimensions there are (1/2)n(n-1) coordinate planes.


I_ ~XXI I











Other Canonical Coordinates


In finding canonical sets of coordinates and moment a wide choice

of momenta,even when the n moment are selected from amongst the con-

stants of the motion,is available. The final decision will depend upon

the particular application desired. However, there are some simple

linear combinations which are often sufficiently useful, to make their

display worthwhile.

As a first selection for the canonical moment consider the follow-
n
ing:
P1 = H =) ala (2.77)
i=2


j-1 n

Pj a= a aia (j = 2, ...n), (2.78)
i=l i=j

where P. is the energy difference between the first (j-l) coordinates
J
and the last n-(j-l) coordinates. The coordinates conjugate to these

moment must then have the form


1 ala
S= In n (2.79)
Q1 = ala n


I a la
Q. = In a laj (j = 2, ...n) (2.80)
J 4i a. a


In terms of the parameters used in the Hopf mapping these become


H = X2/2


(2.81)












P2 (X2/2) cos 0


Pj =


(X2/2) [cos2 T


j-1


+i.
i=2


j-1
= (%2/2) [cos a + Efsin2 0/2]
i=2


2 2 2 ]
eisl n ]


n
E Sin T -
i=j


(j = 3, ...n) (2.83)


Ql = 1/2(p on)


Q2 = 1/2(p 02)


Qj = 1/2(ao. 0.)
J J-1 J


(j = 3, ...n).


One might also consider taking the total energy and the energy in the

last n-(j-l) coordinates for the canonical moment,


n
PI = H = aia'
i=l


(2.87)


n
PJ = a;a-


(j = 2, ...n)


with the corresponding coordinates


1 a.a.
S naj a2+
2: aa+


(j = 1, ...n-1)


1 a
n
S= In -
2i a,


and


(2.84)


(2.85)


(2.86)


(2.88)


(2.89)


(2.90)


(2.82)












which in terms of the Hopf parameters become

H = X2/2


n
P = (2/2)Z ? sin2 1
i=j


Q = p -02


Qj = oj aj+


Qn= on


(j = 2, ...n)


(j = 2, .. .n-1)


Finally, one might consider using the energy and(n-l) successive

energy differences,


n
PI = H aiaYi
i=l


P = -.a j = (2, ...n)



The corresponding coordinates are

S a1a2 ..an
Q In n
2ni ala2 ...an


I k-1 n-k+1 n
Qk In ( (1) ) ( i
2ni j=l aj j=k


k-I

a
J


(2.96)


(2.97)


(2.98)


(2.99)


k = (2, ...n).


(2.91)



(2.92)


(2.93)


(2.94)


(2.95)












In terms of the Hopf coordinates


P, = H = X2/2 (2.100)


P2 = (2/2)(cos2 T E2 sin2T) (2.101)


P = (\2/2) sin2 -E e:) (j=3 ...n) (2.102)

n
Q = (l/n)(p + oi) (2.103)
i=2

and J-

Qj= (/n)[(n-j+l)(p + aO)-(j-1) ok] (2.104)
k=2 k=j

In all three cases, the transformation has been such as to give

the action-angle variables. The effect has been to map the orbit into

a point on the surface of a sphere of 2n-l dimensions whose radius is

the energy H. Since the mapping is many to one, one sees that as in

the two dimensional case an entire orbit is mapped into a single point.

The orbital phase may still be recovered by attaching a flag to the

point representing the orbit which will then rotate with constant

angular velocity Q1, since Q, is conjugate to H and hence increases

linearly with time.

The second case also has another interesting property, namely D2

is the Hamiltonian for an isotropic oscillator in (n-1) dimensions,

and the other Di are constants of its motion. Hence one may perform

another Hopf mapping and continue in this manner until one has the 2-

dimensional oscillator again.











In general, any of the 2n-1 independent constants from the n2-1

constants could be chosen as generators, and used as coordinates. It

should be noted that the expressions for the coordinates could just

as well be expressed in terms of an arctangent by using the well known

relation

tan-I x = i n -ix (2.105)
2 l+ix

which would exhibit the similarity between these coordinates and those

obtained by Goshen and Lipkin [23].




The Anisotropic Oscillator in n Dimensions


In units where the mass is unity, the Hamiltonian for the n-

dimensional anisotropic harmonic oscillator is

n
H = (P + xi), (2.106)
i=1

where wi is the frequency in the ith coordinate. The eigenvectors are

all of the form

a = (Pj i w. xj)AF2 (2.107)


a = (Pj + i ~j xj)/-2 (2.108)

with eigenvalues

x. = i J (2.109)
J J

There also exists n2 constants similar to those obtained in the iso-

tropic case, namely









n
H aial
i=1


Di = aia ai+1 aTi+ (i = 1, ...n-1)


(a J a' + ai J a ))
Kij w.-l ui-l

(aiaf)2 (aj ay) 2



i[a. J a. a. J a ']
Lij i 1

(aia ) 2 (aja) 2


(2.110)


(2.111)



(2.112)






(2.113)


Once again this set is closed under Poisson bracket and satisfies

the commutation relations of the generators of SUn*

It should be noted that every normal quadratic Hamiltonian is

equivalent to some anisotropic harmonic oscillator, because they can

be put in the canonical form given in Chapter 1.

For non-normal Hamiltonians such as the free particle

1 -2
H = (2.114)
2m

there is no equivalence with the harmonic oscillator. However, symmetry

groups still exist for these cases; for example the above expression is

invariant under rotations. The symmetry of such Hamiltonians has been

discussed in considerable detail by Moshinsky [38] and F.T. Smith [39,

40].











For the Hamlltonian given In Eq. (2.114) (where m 1)

ing set of nine quadratic constants of the motion exists,


H = (P2 + p2 + p2)/2
x y z
2 2


A = P2 P2
A2 z2
=PP

A3 = PXPy


A4 = PxPz


A5 PyPz


L1 = yPz zPy


L2 = zPx xPz


L3 = xPy yPz


The relations between these constants are summarized in the

Poisson bracket table:


the follow-




(2.115)


(2.116)


(2.117)


(2.118)


(2.119)


(2.120)


(2.121)


(2.122)


(2.123)


following











H Al A2 A3 A4 A5 L1 L2 L3


0

0

0

0

0

0

-2A5

-2A4

4A3


0

0

0

0

0

0

4A5

-2A4

2A3


0

0

0

0

0

0

-A3

AI+A2

A5


0

0

0

0

0

0

-A2

A3

-A4


0

2A5

-4A5

-A4

A3

A2

0

-L3

L2


0 0

2A4 -4A3

2A4 -2A3

A5 Al

-AI-A2 -A5

-A3 A4

L3 -L2

0 Li

-Ll 0

(2.124)


These define a group which is the

dimensional rotation group and the


semi-direct product (41] of the three

five dimensional abellan group formed


by the five symmetric second rank tensors. In general, a free particle

in n dimensions will have n2-1 linearly independent constants of which

n(n-l)/2 will be the generators of Rn. The others, n(n+l)/2 -1 In

number, commute among themselves and are the generators of the abellan

group. These constants are second rank tensors formed by taking pro-

ducts of the moment.

The harmonic oscillator Hamiltonlan will be generalized in the

next chapter and consideration given to the problem of a plane harmonic

oscillator in a uniform magnetic field.














CHAPTER 3


THE HARMONIC OSCILLATOR IN A UNIFORM MAGNETIC FIELD


Introduction


In the previous chapter it has been shown how the accidental de-

generacy of the harmonic oscillator may be accounted for by the methods

presented in Chapter 1. Although historically, attention has most often

been focused upon the accidental degeneracies of the harmonic oscillator

[9] and the Kepler problem [4, 51, the problem of the cyclotron motion

of a charged particle in a uniform magnetic field possesses certain

unique features, among which is that it has two linear constants of the

motion. It also has accidental degeneracies of Its own [42]. Its

Hamiltonlan,
H = ( A)2/2m, (3.1)


where, for one choice of gauge


A = (- By, + BoX, 0) (3.2)


Is quadratic.

As shown In the previous chapter, the constants of the motion of

the two dimensional harmonic oscillator K, L, and D arose in a natural

fashion as products of linear elgenfunctions of the operator TH. The

eigenvalues of TH occurred in negative pairs, so that their products,

belonging to the sums of the corresponding eigenvalues, were constants

50











of the motion. Moreover, there was a convenient mapping, the Hopf mapping,

which yielded a set of canonical angular coordinates 4 and T, whose con-

jugate moment were respectively, H and D.

In the present instance, one finds by expanding Eq. (3.1) and using

the gauge In Eq. (3.2) that

1 e28~2 eBo
H (P2 2) o (x2 + y2) + (yx xPy). (3.3)
2m 8mc2 2mc


Effectively this is the sum of a harmonic oscillator Hamiltonlan and a

term proportional to the angular momentum.

There is one main difference between the problem of cyclotron motion

and the harmonic oscillator. The representation of the harmonic oscil-

lator Hamlltonian, applied to the homogeneous space H(1) has negative

pairs of non-zero elgenvalues. All constants of the motion are generated

by products of the corresponding eigenfunctions, for which the eigenvalue

sum Is zero. Thus the lowest order constants of the motion are quadratic,

belonging to H(2), and all others are linear combinations of these pro-

ducts. With the cyclotron Hamiltonlan there are zero eigenvalues, and

therefore there are elements of H(l) which are constants of the motion.

Whereas the Poisson bracket of two quadratic constants Is quadratic, the

Polsson bracket of two linear constants is a scalar. A rather different

kind of symmetry group results in the two cases; for the harmonic oscil-

lator one finds the unitary unimodular group, while for cyclotron motion

one obtains a group generated by the harmonic oscillator ladder operators.











Classical Zeeman Effect for the Harmonic Oscillator


It is somewhat more instructive to consider a plane isotropic har-

monic oscillator in a uniform magnetic field rather than the cyclotron

motion exclusively. Its Hamiltonian,

(P-e 2 1 22
H = (- A) + m r



= I (p2 + p2) + (w + ()(x2 + y2) + (yx xP)
Tm x 2 '
(3.4)

Is also quadratic. In Eq. (3.4) w is the Larmor frequency eB/2mc and

o is the natural frequency of the oscillator. This Hamiltonian reduces,

in the limit as w -+0 (B -.0) to that of the plane harmonic oscillator,

while as (. -i0 It reduces to that of cyclotron motion.

The matrix representation of this Hamlltonian considered as an

operator under Poisson bracket and denoted by TH Is


0 W m((w2 + 2) 0

-( 0 0 m(w2 +
TH 1 0 0
m
0 -J 0 (3.5)
m

where the basis is composed of the monomials (x, y, Px' Py). The

elgenvalues and elgenvectors of TH are










Elgenvector Eigenvalue

u i[(w2 + w2)1/2 +

u* -il[(2 )12 + / ]


v 1[(w2 + 2)1/2 w]

v* -i[(12 + 12 ] (3.6)

By defining

r = x ly (3.7)

SP = Px iPy (3.8)

the eigenvectors u and v can be written as follows:


u [m(w2 + w2)]1/2 r. + P+ (3.9)


v = [m(w2 + w2)]1/2 r + P- (3.10)

where u* and v* are simply the complex conjugates of u and v respec-

tively. These four elgenfunctions satisfy the following relation


(u*, u) = (v*, v) = 41(2 + w2)1/2 (3.11)

The constants of the motion will be products of eigenfunctlons,

the sum of whose eigenvalues is zero. Hence one establishes the

quantities uu*, vv*, u*Rv, and uRv* as constants of the motion where

R is a number such that


R\I = \2 ,


(3.12)










and


\ = (w2 + w2)1/2 + W


\2 ( + U )1/2 .


In order to

following linear


display the symmetry group in a convenient

combinations are taken as the constants of


H = [l uu* + 2 vv*]/4/2 + 02
R-l
K [uRv* + u*Rv]/R1/2(uu*)

R-l
L = i[uRv* u*Rv]/R1/2 (uu*)f


D = [uu* Rvv*]/R ,


where H has been written in accordance with

quantities satisfy the following relations:


Eq. (1.29). These four


TH(K) TH(L) = TH(D) = 0


TK(L) = aD


TL(D) = aK


TD(K) = CZL,


(3.19)


(3.20)


(3.21)


(3.22)


where
where a 8/2+ j (3.23)


The first of these equations is simply a statement of the fact that K,

L, and D are constants of the motion, while the latter three equations

show that the symmetry group of the system is SU2.


(3.13)


(3.14)


form the

the motion:


(3.15)


(3.16)


(3.17)


(3.18)











If in Eq. (3.12) R is a rational number, then the classical system

has bounded closed orbits and a quantum mechanical analog exists for

the operators K and L. However if R happens to be irrational, then the

orbits are space filling.




Limiting Cases


In taking the limit as w -+0, i.e. as the magnetic field is turned

off, Eq. (3.4) becomes the Hamiltonian for the two dimensional Isotropic

harmonic oscillator, the elgenvectors become,


u Nm wo r + mP (3.24)


v -m Wo r" + P" (3.25)
.vm

while the four eigenvalues degenerate Into two, namely iwo, whence

R = 1. Expressed in terms of the new u and v, the four constants are


H (uu* + vv*)/4 (3.26)


K uv* + u*v (3.27)


L i(uv* u*v) (3.28)


D = uu* vv* (3.29)


which are the constants previously obtained for the Isotropic oscil-

lator. The commutation rules for K, L, and D also still hold.











In considering the limit as w~ -~0 one finds that


u =N-m u r*+ + P+



v = -m m r" + P~
J^m


and


(3.30)



(3.31)


The eigenvalues of u and u* approach 21w in this limit while those

belonging to v and v* both approach zero. Hence, in order to satisfy

Eq. (3.12) R must also approach zero. The Hamiltonian In this limit

approaches that for pure cyclotron motion as in Eq. (3.3). From the

values of the eigenvalues one immediately has two linear constants of

the motion, v and v*, and one quadratic constant uu*.

The constants can be explicitly derived by considering the com-

mutation rules of K, L, and D in the limit as wo -0.

Rewriting the commutation relations explicitly gives


SuRy* + u*Rv
R- '
Ri/2(uu*)



i uRv* u*Rv
R-l '
R1/2 T) 2


uRv* u*Rv
R-l
R1/2 (uu 2


uu* Rvv*

R


uu* Rvv*,(3.32)
,(3.32)

R


= uRv* + u*Rv (3.33)
R-l
Ri/2(uu*) 2


Suu* Rvv*,

R


uRv* + u*Rv
R-L
RI/2(uu ) 2


i uRv* u*Rv (3.34)
R2 R-1(
R1/2(uu ) 2


Multiplying the first of these by R and the latter two by R3/2 and then


and










taking the limit as wo -0 (R -+0), results in the equations

I ((uu*)1/2 (v*+v), (uu*)1/2 (v*-v)) = 8 uu* (3.35)

I ((uu*)1/2 (v*-v), uu*) = 0 (3.36)

(uu*, (uu*)1/2 (v* + v)) = 0 (3.37)

Since uu* Is simply twice the cyclotron Hamiltonlan, it follows from

the last two equations that both the real and imaginary parts of v are

constants of the motion. Dividing Eq. (3.35) by uu* gives


(2 v = 8w (3.38)

Expanding Eq. (3.1) results in

H = (P2 + P)/2m + mw2 (x2 + y2)/2 + (yPx xP ).
x y y ( 3 3 9)
(3.39)

Thus the cyclotron Hamiltonlan splits into two parts, one being the

harmonic oscillator Hamiltonlan Ho and the second proportional to the

Z-component of the angular momentum L. Both of these terms commute

with the total Hamiltonian H = Ho + L. Hence another quadratic con-

stant of the motion Is

D = Ho L (3.40)

For convenience in notation define


S -m -(v-v*) my Px (3.41)


Q Sm (v-v*) mW + Py (3.42)











With these definitions the following commutation rules hold:


(H, D) = (H, S) (H, Q) J 0 (3.43)


(S, Q)} 2mw (3.44)


(D, S)} 2wQ (3.45)


(D, Q) = -2wS (3.46)


These commutation relations coincide with those obtained by Johnson

and Lippmann [42]. These authors have discussed the two constants S

and Q in considerable detail and show that they are simply related to

the location of the center of the circular orbit and to its diameter,

which can readily be seen In the following manner. The canonical

momentum expressed in terms of the mechanical momentum is


S= my + E (3.47)
c

Substituting for the canonical moment in Eqs. (3.41) and (3.42) gives


S = m(2wy vx) (3.48)


Q m(2w + vy) (3.49)


Evaluating S when vx = 0 and Q when vy = 0 gives the center of the

orbit as
(xc, Yc) = (Q/2mw, S/2mw). (3.50)


These constants also determine the diameter of the orbit. Since the

orbit is a circle only one of the constants must be considered. For











example, consider Q. When y takes on its maximum positive value, x

takes on its minimum value and when vy takes on Its maximum negative

value, x is a maximum and hence the diameter d is


d = xmax Xmin m (3.51)
w

Because of the continuum of points available for the center for a

given energy the degeneracy of this problem is infinite.




Canonical Coordinates


As in the case of the plane harmonic oscillator a set of canonical

coordinates can be found such that the Hamiltonian becomes a canonical

momentum. In fact, two moment for the problem are:


S= uu* + X2 vv*
H = (3.52)
4( 2 2 1/2
and

D uu* Rvv* (3.53)
R


while the coordinates conjugate to these moment are


Ql 2[ln + %i In v]/4i Xl,2 (3.54)
u v


Q2 2[In u* X In v]/16i ( 2 + 2)1//2 (3.55)
Q %2 u I v X2 (w +0) (.5


respectively.











A mapping similar to the Hopf mapping may also be performed where


u = [% /2 cos T e"p] (3.56)

and
v = [l/2 sin T e-I ] (3.57)


Under this mapping the moment become


H = k/4 (2 + 2) 1/2 (3.58)


D = %I cos 9 (3.59)

where
O 27 (3.60)

and the coordinates are

QI = Y/2X2 (3.61)
and
and 2 2 1/2
Q2 = */8R(2 + W /2, (3.62)
where
T = Rp + o (3.63)

and
= Rp a (3.64)


Performing the mapping on the other two constants of the motion gives


K = Xi sin 0 cos 0 (3.65)

and
L = Xs sin 0 sin 0. (3.66)


There also exists another set of canonical coordinates which were

originally defined by Goshen and Llpkin [24]. Written In terms of the

cartesian coordinates the moment are taken to be










P = [(p + P2)/2m + m(W2 + 2)(x2 + y2)/2]/(w2 + 2 .67)
q x y 0 0


P = XPy yP .


The corresponding coordinates are


o = tan'
2


-1
q = I tan-1 2
2 (2


[P P /m + m(i2 + w2) xy]
2 2y 2 2 (3.69)
(Px + P2)/2m + m(u2 + o2)(x + y2)/2


(o2 + 2)1/2 (xP + ypy)

P)/2- m( T + )(x +


In terms of these variables the Hamiltonlan has the particularly simple

form


H = (WZ + U) I/2 Wp0


(3.71)


from which It follows that both Pq and PO are constant In time.




Rotating Coordinates


The problem of the plane harmonic oscillator in a uniform magnetic

field has a certain uniqueness when viewed from a rotating coordinate

system. However, the problem will first be solved In plane polar

coordinates. An extremely lucid description and tabulation of these

orbits has been given by E.R. Harrison [431.

Assuming the direction of the magnetic field to be in the negative

Z direction, the Lagranglan is


and


(3.68)


and


2)/2--.(3.70)
y2)/2


I











m 112
L = (r2 + r262) m r2 mwr2b
2 2

where the following gauge has been chosen,


Ar = Az = 0

and
AO = Bor


The moment conjugate to r and 6 are


Pr = mr

and
SP = mr2( w).


Since L Is cyclic in ,. Pn is a constant of the motion.


(3.72)





(3.73)


(3.74)





(3.75)


(3.76)


The Hamiltonian


is
H Prr

2
= (Po


and the equations of


+ P,9 L


+ Pg/r2)/2m +


motion are:
2
P PO
mr3


PO = 0


= Pr/m


6- +P
mr


In general, the effect of imposing


m(w2 + w2)r2/2 + uPo


S2 2
m(W + +w)r


a uniform magnetic field on a system


(3.77)


(3.78)



(3.79)


(3.80)


(3.81)











with a central potential Is to add two terms to the Hamiltonian, namely

a harmonic oscillator potential which Is often neglected for small

fields [44], and a term proportional to Pg, the angular momentum.

The solutions for the orbit equations are:



0 0. a sin-1 br2-2a2 )+ sin-1 [ 2W2r2-b )
21aI r2[b2_4a2W21/2 2W [b24a2W2]1/2



a br2 -22 w 2W2r2 -b
--- sin-' sinl
21a r[b2-24a2 1/2 2W [b24aW2 1/2

(3.82)

where 2 22 2
b = 2 2r + (3.83)
r2
0

a = Pe/m (3.84)


W = ( +2 )/2 (3.85)


The subscripts on the coordinates and velocities denote Initial

values. These orbits are plotted In Fig.3.1 through 3.4. In all cases

units have been chosen such that m q c = 1. Fig. 3.1 and 3.2

show the high and low field orbits respectively, for the same set of

Initial conditions and wo. Fig. 3.3 and 3.4 show the orbits for a

fixed magnetic field, but for different values of the initial tan-

gential velocity. In general the orbit will be a processing ellipse

for PO > 0 and will be a hypotrachold similar to Fig. 3.1 for Pg < 0.





64








Parameters Initial Conditions

= 63/4 r = 1 r 0
o o
w 4 =0 6 =1
o o o

















0=0










e 1


Fig. 3.1 Particle orbit in a strong uniform magnetic field
and harmonic oscillator potential.

























































































E

U 3 0
0. 3


O



0
. I


- 0

U I

0 0
IL C


O


0
0.


41




u
0







-o
u
















0
U




























4-1














0
L-
0
























L.
t-
C
0,
(U











u-
0


6I
E


0

















U-
*-i













0->























L-'

H

0


u >







4-
c -












0U
EO























L
0 >
-o -
Cr0
10








o .
U-



-c

*-J
0





E -










- C











-CO
U-











LU7
(gIn


























I
0
o<


Cu






U



4-


0)









4- E
C
u-
CU

E -

04
Sc4








- 0




.00-

OL.

0&

- i.
U-








4 O
w
C)
3 HI
OUJ
IU *







u-0

^v
(U <-



- m










In a rotating coordinate system defined by


S= 0 wt (3.86)


S= r (3.87)


where w is the Larmor frequency, the Lagrangian is

m -2 -2 -2 2 m-(o -2
L = [r + r ( r (3.88)
2 2

The canonical moment in the rotating coordinates system are:


PF = mr (3.89)


P- = m;2 (3.90)
e

L is also cyclic in 8 and hence P- is a constant. The Hamiltonian in

the rotating coordinate system is


H PrF+ P- 0 L
0

S(Pr2 + P2 )/2m + m(2 + 2)r /2 (3.91)


and hence the equations of motion are



= m(w2 + 2)? (3.92)
r mF3 o

P = 0 (3.93)


r Pp/m (3.94)

P
S= -- (3.95)
m:2











which are the equations of motion for a plane Isotropic harmonic oscil-

lator with force constant m(w2 + w(). In general the transformation of

a Hamiltonian with a central force from a stationary coordinate system

to a rotating coordinate system will subtract a term proportional to PO.

Generalizing then, It can be stated that a Hamiltonian with a central

potential and with a uniform magnetic field present, when viewed from

a rotating coordinate system, has a harmonic oscillator potential added,

that Is

H T + V(r) + 1 mw2 2 (3.96)
2

where T is the kinetic energy In the rotating coordinate system and w

is the rotation frequency of the coordinate system with respect to the

fixed system. In the present case since V(r) is a harmonic oscillator

potential one simply has a harmonic oscillator with a larger force con

stant.

The solution for the orbit defined by the Eqs. (3.92) through

(3.95) is

a b a2 a b2--a2a2
-60 s-i-n-] (n si)- sin-) ( o
21al F2[b2 -4a2 2 2 1/2 2a s 1 2b2-4a 2 2 1/2


(3.97)
where W is defined in Eq. (3.85) and


a = P5/m (3.98)


b a2/F2 + w2 F2 + (3.99)

Equation (3.97) Is the equation for an ellipse, regardless of the











magnitude of w. The orbits corresponding to Fig. 3.1 and 3.2 in a

rotating coordinate system are shown in Fig. 3.5 and 3.6.




Discussion of the Constants of the Motion


Of the four constants of the motion defined in Eqs. (3.15) through

(3.18) one is H, the energy of the system. Another constant is the

angular momentum PO which is a linear combination of H and D


PO = H + R (3.100)
XI2 4(w2 2o)1/2

Since only three of the constants are independent one would like to

discover one other constant, independent of the two above, which has

a physical meaning. This can be done by considering the constants of

the motion in the rotating coordinate system and then transforming back

to the original coordinate system.

In the rotating coordinate system two constants of the motion are

aa* and bb*, where

a = PR i(2 + o)1/2 x (3.101)

and
b -P 1(w2 + ) 2)1/2 (3.102)


and where the mass is taken to be unity. By a simple rotation the axis

can be oriented so that the semi-major axis of the ellipse lies along

x and the semi-minor axis lies along y. In this case then aa* Is pro-
portiona to the maximum value which 2 attains, because since
portional to the maximum value which x attains, because since








71














o 44
0o



u
CC





o
0

o
0
U










E
0
L





-4
-o









3





0,
C







o








r)






L-
tjE









72






o

0





E
U)




4)

O
L


0
0
U






0
'.

lu
r-














E
O











0
L
/I I 8
,-







,10





3
Lo





0
u
-r










a-










aa* P-2 + (w2 + (2) g2
x o


is a constant, it can be evaluated when P. = 0. Similarly bb* Is pro-
x
portional to y max. Alternatively, one can say that aa and bb' give

the boundaries of the orbit i.e. aa* gives the maximum radial distance

the particle can attain and bb* is the minimum radial distance.

These two radii are also constants of the motion In the stationary

system since the transformation only involved the angle. However, it

can explicitly be shown that these are constants.

By transforming from the rotating coordinate system to the station-

ary one aa* can be written in terms of the eigenvectors and their con-

jugates as


aa* =(1/4)(uu* + vv* + uv* e2iWt + u*v e-21t) .


In the stationary system the time rate of change is


(3.104)


d (aa*) = (aa*, H) + (aa*)
dt /t


=(1/4)[-21wuxjve21wt + 21 u*ve-21wt


+21iuv*e2iwt 21iu*ve-21tt] = 0,


(3. 105)


and hence aa* is a constant of the motion in the stationary system also.

A similar proof also shows bb* to be a constant. However aa* and bb*

are not Independent, but are related to one another through H and D.

Even though a good physical interpretation cannot be given to the

constants K, L, and D their effect on the orbit under Poisson bracket

can be calculated i.e., the infinitesimal change each produces.


(3.103)










Before calculating the effects of the constants on the orbit, it

is desirable to write the orbit equation in terms of the constants of

the motion and the time. This can be done by defining a complex vector


r = x + ly (3.106)


which can be written In terms of the eigenvectors by inverting Eqs.

(3.9) and (3.10)
r = (u + v*)/2c (3.107)

where
c = [m(w2 + W21/2. (3.108)


Employing the polar forms of both the eigenvectors in Eqs. (3.56) and

(3.57) and of the constants in Eqs. (3.59), (3.65) and (3.66), u and v*

may be expressed in terms of the constants as


u = (R/2)1/2 [(K2 + L2 + D2)1/2 + D]1/2 e-i' (3.109)


v* = (R/2)1/2 ((K2 + L2 + D2)1/2 D]1/2 ei' (3.110)


However from Eqs. (3.63) and (3.64)

p = (1/2R)(Y + P) (3.111)

and
o = (1/2)(T P) (3.112)

Hence

u = (R/2)1/2 [(K2 + L2 + D2)1/2 + D]1/2 e-1i/2R e-I~/2R
(3.113)
and


S (R /2 [(2 2 2)1/2 1/2 -1I/2 1T/2
v =(11/2) [(K + L + D ) DI e e


(3.114)










From Eqs. (3.65) and (3.66) It is apparent that


0 = tan' L (3.115)
K

The time dependence is in Y since Y1/2\2 in Eq. (3.61) Is the coordinate

canonical to H and hence from


d/dt (~/2%2) = ({/2%2, H) = 1 (3.116)

it follows that
Y = 2%2(t to) (3.117)

or
S=- 2R l(t to) (3.118)


Substituting Eqs. (3.115), (3.117), and (3.118) Into Eqs. (3.113) and

(3.114) and then substituting these Into Eq. (3.107)gives


r = 1- (.E)1/2([(K2 + L2 + D2)1/2+D]1/2e(-i/2R)tan-1 (L/K) e-i'l(t-to)
2c 2


+ [(K2 + L2+ D2 1/2-D /2(-i/2)tan- (L/K) ei.2(t-to))

= r + r (3.119)

where
r 1 ()/2 [(K2 + L2 + D2) 1/2D1/2 (-i/2R)tan1 (L/K) e-ll(t-to)
S2c 2
(3. 120)
and

+ = ( R /2[(K2 + L2 + D2)1/2-D]1/2e(-i/2)tan-' (L/K) eIX2(t-t).
+ 2c 2
(3.121)
Equation (3.119) expresses the orbit equation as the sum of two contra

rotating vectors, whose frequencies of rotation are %1 and X2, and










whose amplitudes are functions of the constants of the motion.

The Poisson bracket of the constants K, L, and D with r are:



(K, T") = L I KD
2 [(K2 + L2 + D2)1/2_D] K2 + L2


L L IKD -
2 [(K2 + L2 + D2 1/2+D] R(K2 + L2) (3.122)



S7 K 1LD
L, ) = [ 2 2 K +ILD
2 [(K2 + L2 + D2 -D/22 K2 + L


a K ILD
2 2 2 1/2 2 72 .
2 [(K + L + D ) +D] R(K2 + L )
(3.123)

i(, la ia-
(D, 1J = r-.. r+ (3.124)


where a is a constant defined in Eq. (3.23).

The infinitesimal change induced by the constants is that each

changes both the amplitude and phase of r+ and r_, while preserving

the sum of the squares of their sum and difference. If the orbit were

an ellipse, this last statement would be equivalent to stating that the

sum of the squares of the semi axes of the ellipse is a constant. The

proof of the statement goes as follows:

Before the infinitesimal change the quantity is


Ir+ + r I2 + r+ r12 = 2(r +12 + Ir_12)


(3.125)










and after the change it is

jr+(1 + E) + r_(l + 8) 2 + r( + r ) r.(1 6) 2

2[r 12 + jr 12 + (E + *) Ir |2 + (5 + 8*) r| 2] (3.126)

where second order terms have been Ignored. The difference between the
two terms is
2[(E + E*) Ir+12 + (8 + 5*) IrI2]

which is zero In all three cases. For example, if the infinitesimal

change is induced by K, one has for the difference

2[(e + E*)Ir+12 + (5 + 8*)Ir_ 2]


2a 2L jrj2 2L |r-12
2 [(K2 + L2 + D2)1/2-D [(K2 + L2 + D2) /2+D]


2aR L[(K2 + L2 + D2)1/2-D L[(K2 + L2 + D2)1/2+Di
8c [(K2 + L2 + D2)1/2-D] [(K2 + L2 + D2) /2+D]

0. (3.127)



Gauge Transformations in Uniform Magnetic Fields

in uniform magnetic fields and in the absence of other external
potentials the following fact concerning a charged particle in this

field Is true: The change in momentum In going from one point to

another Is Independent of the path taken between the points. The











proof Is as follows: Let P denote the canonical momentum, A the mechani-

cal momentum, B the uniform field, and A the vector potential such that


B Vx A (3.128)

and since B is uniform

A B x r (3.129)
2
Since
d e V x (3.130)
dt c

and
v dr (3.131)
dt

one has -
rI r2
f d f f (- x B) dt
c dt


F2
= .f dr x

Trl

T2
2e f dA (3.132)
c -
rI


Hence
S(3.133)
c
and
aP - (3.134)
c

Equation (3.134) states that the difference in moment between any two

points is proportional to the difference in the vector potential evalu-

ated at these points. From this it Is Inferred that the choice of a

certain gauge Is equivalent to picking the zero of momentum.











One can also see why a translation of coordinates must be accompanied

by a gauge transformation. When the translation occurs the zero of momen-

tum changes and hence this change must be subtracted from A to give the

the same zero of momentum. However, the change In A is simply a gauge

transformation.

In order to show that a gauge transformation can induce a translation,

first consider the case of a free particle In a uniform field with the

following gauge A B (-y, x, 0) (3.135)
2

In units where m and w eBo/2mc are one, the constants of the motion are


and


Using the fact that


P y a


P + x -B .
p +x-1


P v+ ,
c


(3.136)


(3.137)


(3.138)


the location of the center of the orbit is found to be


(Xc, yc) 2 (2, 2)


(3.139)


However with the gauge


B
A = -2 ( -(X + y), + x, 0),
2


(3.140)


even though the two constants are the same, one finds the center of the

orbit to be


(xc yc) ( __2 a3- X )
2 2


(3.141)











Hence the center of the orbit has been translated -1t/2 units in the x

direction and X/2 units in y.

The center of the cyclotron orbit may also be rotated, however

this is not accomplished by a simple gauge transformation. Instead

one must rotate the vector A and also rotate the coordinate system.

For the vector potential defined in Eq. (3.135) the center of the orbit

is given in Eq. (3.139). A rotation of the vector field defined by 1

gives
cos 6 sin 0 0 -y
B,
AR = -sin 0 cos O 0 x
2
0 0 1 0

B
= ( -y cos 6 + x sin 0, y sin 0 + x cos 6, 0). (3.142)
2

The rotation of the coordinates defined by


x U x' cos 6 y' sin 6 (3.143)

and
y x' sin 0 + y' cos 6 (3.144)


transforms AR into A' where

B ,
A' = (-y', x', 0) (3.145)


Hence, the center of the orbit is


(xc ) ( -, ) (3.146)

and


B B cos 0 + a- sin 0
2 2 2


(3.147)











a B' a'
S -- sin 0 +- cos 0, (3.148)


which shows that the center of the orbit has been rotated with respect

to the original coordinate system, while the diameter of the orbit has

remained unchanged.

The vector field can also be rotated without rotating the coordi-

nates and still preserve the curl if at the same time the vector is

dilated by a factor of sec 6. The new vector potential is

Bo
A = sec 6 (-y cos e + x sin 6, y sin 6 + x cos 0, 0).
2 (3.149)


The transformation leaves the center of the orbit invariant and in fact,

Is equivalent to a gauge transformation where the vector AD


B
AD = tan O(x, y, 0), (3.150)


whose curl is zero, has been added to the original vector potential A.













CHAPTER 4


THE KEPLER PROBLEM


The Two Dimensional Case in Polar Coordinates


The Hamiltonian for the two dimensional Kepler problem in polar

coordinates Is
H = (1/2)(P2 + p2/r2) 1/r (4.1)


where the mass and force constant have been chosen equal to unity.

The corresponding Hamilton-Jacobi equation is

(S )2 + (,)2 2(W+ ) = 0. (4.2)
or r2 ao r

where W is the total energy and


S(r, 4) = Sr(r) + S(~Q) (4.3)


Hence
p S (4.4)
Pr 7r

and
p Bs
P ; (4.5)

Since the partial differential equation (4.2) has no P dependence

bS/ba Pp Is a constant and is, in fact, the angular momentum of the

system. Solving Eq. (4.2) for 6S/or = Pr one finds

2_2 1/2
= Pr = [2W + 2 (46)
dr r r2











The action variables Jr and JO are given by the integrals


and


(4.7)


(4.8)


Jr P rdr


J, = 5 PVd$ .


Equation (4.8) can be Integrated immediately since the integral over

one cycle is simply an Integral from 0 to 2A, hence


J = 2P .


(4.9)


The Integral in Eq. (4.7) can

as described in Born [35] and


be evaluated by a contour Integration

the result is


2*
Jr = 1/2 J
[-2W]


(4.10)


The angle variables wr and wo

jugate to Jr and Jo are given


which are canonical coordinates and con-

by


6S Pr
Wr = j" dr
r r
and

sS Pr 3P$
r = f dr + JZ- d


Upon performing the integrations


wr = (u E sin u)/2xr


and


w =- [u 6 sin u tanl (y/x) + <]/2n,


(4.11)



(4.12)


(4.13)


(4.14)











where the elliptic anomaly u is defined through


r = a(l E cos u),


a being the semi-major axis and E being the eccentricity of


a = W/2

2 2


By choosing o to be the azimuth of the perihelion,


x r cos ( 0 0)


y = r sin ( () ,


(4.15)


the ellipse,


(4.16)


(4.17)


and Eq. (4.14) reduces to

wo = (u E sin u + 0 )/2x (4.20)


Solving Eq. (4.10) for the energy in terms of Jr and J

22
w = 2(4.21)
(Jr + 2

and hence the frequencies vr and v are degenerate i.e.,


V v & = (4.22)

T* a a-b
Thus the quantities aa bb ab, and ab are constants of the motion

where J 1/2 2riw
a = (-) e r (4.23)











and b = ()1 e2nw (4.24)
2n

These quantities satisfy the following Polsson bracket relations


(a, a*) (b, b*) i (4.25)


where one uses the fact that the J's and w's form a set of canonical

coordinates and moment.

Taking the real and imaginary parts of these constants and the sum

and difference of the real ones gives the following four constants of

the motion,

H = aa* + bb* (4.26)

D = aa* bb* (4.27)

K = ab* + a*b (4.28)

L = i(ab* + a*b). (4.29)


Of these constants H commutes with the other three and is a function of

the Hamiltonian, namely, it is inversely proportional to the square

root of the energy. The three constants K, D, and L satisfy the com-

mutation relations of the generators of the group SU2 or R3.


(K, D) 2L (4.30)

(D, L) 2K (4.31)

(L, K) = 20 (4.32)

Although they have no simple Interpretation as constants of the motion,

one can make a mapping of these constants which preserves the commuta-

tion rules. The new constants can be interpreted as the angular











momentum and the two components of the Runge vector [Il].

is
K -K' = K(2 + H+D)1/2
H-D


L -+L' = L(2 + H+D)1/2
H-D


D -D' = H D


These constants satisfy the rules


(K', D')

(D', L')

(L', K'}


= -2L'

= -2K'

= -2D'


Otherwise they are a direct consequence of

K, L, D, so that this mapping is valid for

must be made of the identity


H2 = K2 + L2 + D2


the commutation rules for

any SU2 group where use


(4.39)


K' is essentially the x component of the vector P where


P = R --2E = (-e//-2E) Cos 0
sin 0
0


(4.40)


and R is the Runge vector, while L' is the y component and D' is the

angular momentum.


The mapping


(4.33)



(4.34)



(4.35)


(4.36)

(4.37)

(4.38)











The Kepler Problem in Three Dimensions


Using the same units as in the previous case, the Hamiltonian for

the Kepler problem in three dimensions is


2 2
1 2 i' P
H = (Pr + -- ) -- (4.41)
2 r2 r2 sin2 0 r


Setting up the Hamilton-Jacobi equation and separating gives


P = S= a = constant (4.42)


S a 1/2

0 sin2 )
and
S 2 1a /2
Pr (2W + (4.44)
ar r r2


where a9 and aO are separation constants, aO being the magnitude of

the Z component of the angular momentum and a0 the magnitude of the

total angular momentum. Since the potential is central the orbit is

planar and the normal to the plane is necessarily parallel to the

angular momentum. Denoting by B the dihedral angle which the orbital

plane makes with the x-y plane one can deduce that


cos B - (4.45)
a0

Solving for the action variables gives


JO = 2r0( = 2 nP


(4.46)




88




Je 2(C(a ca) (4.47)


Jr JO JO (4.48)


and from Eq. (4.48) It follows that

2

(Jr + Je + j)2 (4.49)


In the plane of the orbit rectangular coordinates 4 and v can be

introduced, where the 4 axis is the major axis of the ellipse and the

origin coincides with the center of force and the origin of the polar

system. In this system

4 = r cos (T To) (4.50)


v = r sin (Y To), (4.51)


where Y is an angle measured from the 4 axis to the particle and To

is the angle to the perihelion.

The angle variables are

oP
wr Pr dr = (u e sin u)/2n (4.52)
6Jr


we J dr + j dJ
PJO r r


= (u e sin u tan-1 V + sln-1 cos O)/2, (4.53)
p. sin B










w= -dr + f dO + f- dP


= (u E sin u tan-1 V + sin-1 Cs sin-' cot + +)/2
i. sin B tan B
(4.54)

where u is defined as in Eq. (4.15) and E as in Eq. (4.17). Using Eqs.

(4.50) and (4.51) It follows that


tan-1 v = T o "(4.55)



It follows from Fig. 4.1 that sin-1 cot 6 is the projection of the
tan B
angle T onto the plane in which 0 is measured I.e., the x-y plane, which

shall be denoted --. It also follows that sin-1 cos 0 Is the angle T,
sin B
where the line of nodes has been chosen to coincide with the major axis of

the ellipse. Hence, the line of nodes makes an angle (0 -) with the

x-axis. Rewriting Eqs. (4.53) and (4.54) results in the following set

of equations,

we = (u e sin u + To)/2i (4.56)


we = (u e sin u + \o + ( ^ -)/2n (4.57)

By defining

a = ()/2 e2iwr (4.58)
2ir


b (Je)1/2 e2niv (4.59)
2n

J= 1/2 27iw
e (4.60)
2c



















































Fig. 4.1 Three dimensional coordinate system for
the Kepler problem.










one finds the following nine constants of the motion: aa*, bb*, cc*,

ab a b, bc b c, ac and a c. These constants are the generators

of the group SU3. The constant angles B, To and 4 - and also the sum

"-+ To can be expressed In terms of the constants,


sin Yo M i(ab* a*b) (4.61)
2(aa*bb*)1/2

ab* + a*b
cos Yo + 1/2 (4.62)
2(aa bb*)

sin( -') = i(bc* b*c) (4.63)
2(bb*cc*)1/2


cos(4 -- (bc + b c) (4.64)
2(bb*cc*)1/2


sin( + ) = i(ac* ac) (4.65)
2(aa*cc*)1/2


cos( - + ) ac* + a*c(4.66)
2(aa*cc*)1/2 66
and
cos B = cc (4.67)
bb* + cc*

Generally, the rotation group In four dimensions R4, is taken to

be the symmetry group of the Kepler problem. Its generators are the

three components of the angular momentum vector L and the three compo-

nents of the normalized Runge vector P which Is defined in terms of

in Eq. (4.40). The rectangular components of P are









cos Yo cos (0

P E -cos Yo sin (0

sin B sin To


) cos B sin yo sin (0 )

) cos B sin T' cos ( --)


(4.68)


For L one has


sin B sin ( -)

sin B cos (P ---)

cos B


Using Eqs. (4.61) through (4.67) one can express

in terms of the generators of SU3 as follows:


S(aa* + 2bb* + 2cc)1/2 [(ab* + a*b)(bc*
4bb*(cc*)/2(bb* + cc*)


the generators of R4




+ b*c)(bb* + cc*)


+ (ab* a*b)(bc* b*c) cc*]


(4.70)


p -l(aa* + 2bb* + 2cc*)1/2
S4bb*(cc*)I/2(bb* + cc*)


[(ab* + a*b)(bc* b*c)(bb* + cc*)


+ (ab* a*b)(bc* + b*c) cc*]


i(aa* + 2bb* + 2cc*)1/2
z 2(bb* + cc*)



Lx = i(bc* b*c)(bb* + 2cc*)
2(cc*)1/2



L (bc* + b*c)(bb* + 2cc)l1
y 2(cc*)1/2


(bb* + 2cc*) /2(ab* a*b)


1/2


JO +
2x


(4.69)


(4.71)



(4.72)



(4.73)



(4.74)










and Lz = cc* (4.75)


These six quantities satisfy the relations


(Li, P) - Eij Pk (4.76)


(Li, Lj} Eljk Lk (4.77)

(Pi' Pj) ijk Lk (4.78)

which define R4, once again directly from the commutation rules for SU3.

The process can be inverted and the generators of SU3 can be ex-

pressed in terms of the components of P and L. These are

aa* (L2 + L2)1/2- L (4.79)
S x-2 y


bb* = (L2 + L2)1/2 (4.80)

cc* L (4.81)


ab* (L2 + L2)/2- L1/22 + L2/2(cos o I sin )

(4.82)
bc* (Lz(L2 + L2)1/21/2 (cos () 1 sin (0 ---)) (4.83)
z x y


ac* [ -(L2 + L2)l/2-Lz]/2Lz /2[cos( ~ + Y )- sin(4 + )]
=2W x y z 1
(4.84)
where L
4 ^= tan" -x (4.85)
Ly













and


(4.86)


- t 1 PzLy(L + L2 + L2)1/2
SP(L2 + L2) + PzLxL


2 2 1/2
S tan1 (L + L)
B = tan-'VX^^ -


(4.87)


The Two Dimensional Kepler Problem in Parabolic Coordinates


in parabolic coordinates, defined by


x =L v (4


y = (L2 v2)/2 (4


and where m = e = 1, the Kepler Hamiltonian is

P2 + p2 4
H = 0 (4
2(42 + 2)
From the Hamilton-Jacobi equation


P[ = (2wL2 + 2a1)1/2 (4


PV = (2Wv2 + 2a2)1/2 (4

where
al + a2 = 2. (4


The action variables defined by

Jl = Pidq (4

give 2nal
= (4
GL -2


.88)


.89)


.90)


.91)


.92)


.93)


.94)


.95)




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