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 CONTENTSContinued
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 nonrelativistic Kepler problem, whose
quantum mechanical analogue is the problem of a oneelectron 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 nonrelativistic and
relativistic cases. In the nonrelativistic [251 case he has obtained
relations for the representations of R4 exploiting the fact that R4 is
locally isomorphic to R3 x R3. The nonrelativistic 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 nonrelativistic 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+n2), 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 f1. Similarly for i < 0 the dual vector is taken to
be f .. Thus the dual basis to
I
(1.8a)
T = (fn, ... fli fi .... fn)
is then
r = (f ... f1 .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 selfdual 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+n2). 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 = gi 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
(Xi + Xi) fgi, gi) = 0, (1.18)
and hence, i. = i since (g., gi) 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 = Cii gigi' 9gk
i=l
n
= ) C [g (9g 9, + 9i 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 gk' gk)
(1.24)
indices
(1.25)
h =X Ci,i gigi.
and hence
C (1.26)
cII = gi gi)
The final conclusion is that
h = g i9i (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 gkgm, (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 gi' gj} + gi gi' gj)]
i=l
n
+ Ck,m [gk9m' 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 = ' gigi (1.30)
(9i, 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 nonnormal 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
Ck,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= ggi+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) = fi, (1.37)
or the time reversal mapping T(fi)
T(fi) fi
i > 0 (1.38)
T(fi) = fi
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 +gi9i (1.40)
2(9_ gi
n
i=1
P gig; g ;g ;
=1 2(gi, gi)
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) = gk
TR(g = (R, gk = k
gk
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 gk k
TR(TT(gk)) = gk = 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 = (Xeip cos T)A/2 (2.16)
b = (\eio 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(OP) 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(op) 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 oneheaded 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 twocomponent spinors.
The Anisotropic Harmonic
Oscillator in Two Dimensions
The anisotropic oscillator, both with commensurable frequencies
and with noncommensurable 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 noncommensurable 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*)(w1)/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 orbita Lissajous figure in the xy planewould 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 nonalgebraic integrals can exist, and as would be expected, are
associated with spacefilling motion.
Another point regards the separability of the HamiltonJacobi
equation in several coordinate systems. It is pointed out that if a
HamiltonJacobi 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 spacefilling 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 spacefilling 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 2n1 independent constants of the motion including
2
the Hamiltonian are expected to be found and not n2. Hence there should
exist n2(2n1) = (n1)2 functional relationships among the n2 constants.
As a consequence, the following set of 2nl constants have been chosen
for the Hopf mapping:
the (n1) constants Klj where j = 2, ...n,
the (n1) 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
2n
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 (n1)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)eij (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 nI complex dimensions. Regarding
the space as having 2n2 real dimensions and performing an inverse
stereographic projection a real space of dimension 2n1 is obtained.
As in the case of the two dimensional oscillator the sphere of
2n1 dimensions is considered tangent to the 2n2 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 northsouth diameter projected from the 2n2 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 2nl coordi
nates in this space. As in the case for n = 2 these 2n1 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(n1) 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
j1 n
Pj a= a aia (j = 2, ...n), (2.78)
i=l i=j
where P. is the energy difference between the first (jl) coordinates
J
and the last n(jl) 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
j1
+i.
i=2
j1
= (%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 J1 J
(j = 3, ...n).
One might also consider taking the total energy and the energy in the
last n(jl) 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, ...n1)
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, .. .n1)
Finally, one might consider using the energy and(nl) 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 k1 nk+1 n
Qk In ( (1) ) ( i
2ni j=l aj j=k
kI
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)[(nj+l)(p + aO)(j1) ok] (2.104)
k=2 k=j
In all three cases, the transformation has been such as to give
the actionangle variables. The effect has been to map the orbit into
a point on the surface of a sphere of 2nl 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 (n1) 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 2n1 independent constants from the n21
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
tanI 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, ...n1)
(a J a' + ai J a ))
Kij w.l uil
(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 nonnormal 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
AIA2 A5
A3 A4
L3 L2
0 Li
Ll 0
(2.124)
These define a group which is the
dimensional rotation group and the
semidirect 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 n21 linearly independent constants of which
n(nl)/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 nonzero 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,
(Pe 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
Rl
K [uRv* + u*Rv]/R1/2(uu*)
Rl
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 =Nm 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
Rl '
R1/2 T) 2
uRv* u*Rv
Rl
R1/2 (uu 2
uu* Rvv*
R
uu* Rvv*,(3.32)
,(3.32)
R
= uRv* + u*Rv (3.33)
Rl
Ri/2(uu*) 2
Suu* Rvv*,
R
uRv* + u*Rv
RL
RI/2(uu ) 2
i uRv* u*Rv (3.34)
R2 R1(
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
Zcomponent 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 (vv*) my Px (3.41)
Q Sm (vv*) 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 eI ] (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 tan1 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 sin1 br22a2 )+ sin1 [ 2W2r2b )
21aI r2[b2_4a2W21/2 2W [b24a2W2]1/2
a br2 22 w 2W2r2 b
 sin' sinl
21a r[b224a2 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
41
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 *
u0
^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 b2a2a2
60 sin] (n si) sin) ( o
21al F2[b2 4a2 2 2 1/2 2a s 1 2b24a 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 semimajor axis of the ellipse lies along
x and the semiminor 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* P2 + (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 e21t) .
In the stationary system the time rate of change is
(3.104)
d (aa*) = (aa*, H) + (aa*)
dt /t
=(1/4)[21wuxjve21wt + 21 u*ve21wt
+21iuv*e2iwt 21iu*ve21tt] = 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 ei' (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 e1i/2R eI~/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)tan1 (L/K) ei'l(tto)
2c 2
+ [(K2 + L2+ D2 1/2D /2(i/2)tan (L/K) ei.2(tto))
= r + r (3.119)
where
r 1 ()/2 [(K2 + L2 + D2) 1/2D1/2 (i/2R)tan1 (L/K) ell(tto)
S2c 2
(3. 120)
and
+ = ( R /2[(K2 + L2 + D2)1/2D]1/2e(i/2)tan' (L/K) eIX2(tt).
+ 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 r12
2 [(K2 + L2 + D2)1/2D [(K2 + L2 + D2) /2+D]
2aR L[(K2 + L2 + D2)1/2D L[(K2 + L2 + D2)1/2+Di
8c [(K2 + L2 + D2)1/2D] [(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 +x1
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 HamiltonJacobi 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 semimajor 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 ab
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
HD
L +L' = L(2 + H+D)1/2
HD
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 HamiltonJacobi 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 xy 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 tan1 V + sln1 cos O)/2, (4.53)
p. sin B
w= dr + f dO + f dP
= (u E sin u tan1 V + sin1 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
tan1 v = T o "(4.55)
It follows from Fig. 4.1 that sin1 cot 6 is the projection of the
tan B
angle T onto the plane in which 0 is measured I.e., the xy plane, which
shall be denoted . It also follows that sin1 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
xaxis. 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 x2 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/2Lz]/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 HamiltonJacobi 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)
