SOVIET PHYSICS JETP

VOLUME 13, NUMBER 6

DECEMBER, 1971

INTERNAL SYMMETRY OF THE MAXWELL "FISH-EYE" PROBLEM AND THE
FOCK \GROUP FOR THE HYDROGEN ATOM
Yu. N. DEMKOV and V. N. OSTROVSKII
Leningrad State University
Submitted September 16, 1970
Zh. Eksp. Teor. Fiz. 60, 2011-2018 (June, 1971)
As a special case of the focusing potentials studied previously, a system is obtained which Maxwellreferred to as the "fish-eye" problem. The wave equation for this problem can be reduced to the
Laplace equation on a four-dimensional sphere. The symmetry group for this problem is, therefore,
the same as that found by Fock for the hydrogen atom in the case of the discrete spectrum. It isshown that the Maxwell potential has focusing properties in the wave sense as well, i.e., when diffraction is rigorously taken into account. The Green function is found, and generalizations of the
fish-eye problem are discussed.
where the point (1; 1, /; 2, ~ 3 , ~ 4 ) lies on a unit sphere in
the four-dimensional space

1. INTRODUCTION
IN a previous paper[ 1J we considered a sphere ofradius R with refractive index
2n'n(r) = 1 (r/R)'+£, = 2Rx I (R' + r'), £, = 2Ry I (R' + r'), s' = 2Rz I (R' + r'),s• = (R' - r') I (R' + r').
(1)

where n(r) = n' when r > R. Let us denote the surfaceof this sphere by S3 • It was shown earlier that all raysleaving a point on S3 in the inward direction werefocused at the point diametrically opposite on S3 • Let
us now suppose that Eq. (1) is valid throughout space.
In the equivalent problem of classical mechanics one
considers the motion of a particle with zero energyll(i.e., such that its velocity at an infinitely distant pointis zero) in a potential
U (r)--'-

2v(R' + r')''(2)

where v > 0 is an arbitrary parameter. In this case,
all the trajectories are circles (or straight lines) intersecting S3 at diametrically opposite points. A
striking feature is that the refractive index given by
Eq. (1) will exactly focus rays leaving any point inspace and not merely points confined to S3 • All theparticle trajectories (or rays) leaving a point r arefound to pass through the point rR 2/r 2 • The image ofan object in such an optical system is the result of its
inversion in the sphere S3 and reflection at the originof coordinates. This transformation is conformal.
The refractive index given by Eq. (1) was firstfound by Maxwell in 1854.[ 2] Maxwell demanded that
all the rays in a spherically symmetric optical systemwere circles. He obtained Eq. (1), and then establishedthe focusing properties of the system which he called
the fish-eye problem.
The reason for the particular properties of the fish
eyewaspointed outby Caratheodory[ 3 J (see also[ 4l). To
demonstrate it, let us perform a stereographic projection of the three-dimensional space onto a four-dimensional sphere S4 of radius R. A point (x, y, z) isthen projected onto the point (RI;h R/;2, R/; 3 , R/;4 ),t>we shall be using the atomic system of units, and will set the particle mass equal to unity.

(3)

The trajectories transform into the major circles ofthe sphere S4, and all these circles which pass througha certain point on S4 intersect at a diametrically opposite point on the four-dimensional sphere, which
explains the focusing phenomenon in the original problem.
The Maxwell fish-eye problem is closely related to
the Coulomb problem. When an electron of energy ~moves on an ellipse in the Coulomb potential Uc ( r)= -Z/r, the family of orbits in momentum space[s,a]is identical with the trajectories which we have considered for the potential given by Eq. (2) ie 1
R'= -28.

(4)

For a monotonic, spherically symmetric potential
U(r) the action function in momentum spacepT(p, p',e)= Jrdp
P'(the integral is evaluated in momentum space over theclassical trajectory joining the points p and p') satisfies the equation(:;.)'+(:;,)'+(:;)'-[r(e-~')]'=o,(5)where r( U) is the inverse of U( r ). [?l In the case of
the Coulomb potential r(U) = 2Z/(U 2 - 2~), Eq. (5) becomes identical with the Hamilton-Jacobi equation(with the momenta replaced by the coordinates) for the
Maxwell problem, provided both Eq. (4) and the following condition are satisfied:
Z'=v.

This will be investigated below for the corresponding
wave problems.
2 lThe motion of the imaging point over a major circle on S4 is, inboth cases, nonuniform and is different for the two problems. Uniformity can be achieved by introducing a new time parameter, andthis is done for the Coulomb problem in [ 7 ].

1083

(6)

Yu. N. DEMKOV and V. N. OSTROVSKII

1084

2. THE WAVE PROBLEM
The fact that the classical trajectories are closed isfound to be related to the additional symmetry of thecorresponding wave problem and the possibility ofseparating the variables in a number of coordinate
systems. The particular feature of our case is that wemust consider the wave problem and the classicalproblem for a fixed energy E = 0:2v]·'•(r)=O[ · _ _!_V'2(R' + r')' "'.

(7)

Accordingly, the variables in Eq. (7) (and in the
Hamilton-Jacobi equation for the classical problem)
can be separated both in the bispherical and toroidal
coordinate systems (as well as in the spherical system)
provided the poles of the coordinate system lie on S3 •These two systems differ by the fact that the Laplace
equation is separable for them, but the Schroedinger
equation with nonzero energy is not. raJ The solution in
the spherical set of coordinates will be given in Sec. 4
for a more general case.
Equation (7) is invariant under inversion in a spherecentered at an arbitrary point b if it cuts S3 on amajor circle (i.e., if its radius is -IR 2 + b2 ). These
inversions form a group. Therefore, if we have a solution 1/J(r) of Eq. (7), the function
R
( r-b
)
lr-bl 'I' (r-b)' (R'+b')+b(8)

will also satisfy Eq. (7). By performing successive inversions in spheres centered on the points x = b andx =i' + Ob on the x axis, we obtain an infinitesimaltransformation. The corresponding infinitesimal operator acting on the wave functions is
A,'= 'j,R-'[-ix + (R'- r')p, + 2x(rp)],where p is the momentum operator. The quantities
Ay and A~ are defined by analogy. The components of
the angular momentum operator L and the operator
A' form the algebra of the 0( 4) group of four-dimensional rotations:found by Fock[oJ for the discrete spectrum of thehydrogen atom.
We note that solutions of Eq. (7), obtained byseparating the variables in bispherical coordinateswith the z axis, are the eigenfunctions of the operators
H, A~2 + Ay2 + L~, and L~, and in toroidal coordinates
they are the eigenfunctions of H, A~2 , and L~, where
the corresponding eigenvalues are the separation constants.
The above representations of the 0( 4) group are
not unitary with respect to the scalar product definedin the usual way, and the operator A' is not Hermitian
because the transformation given by Eq. (8) does notconserve the normalization of the wave function. The
representations are unitary with respect to the scalar
product with weight ( R 2 + r 2 2. The introduction of
the scalar product is equivalent to the reformulation of
the problem, given below. Unitary representations canbe obtained if we consider a new formulation of the
problem, which is analogous to the Sturm problem forthe hydrogen atom.( 10 J If we use the wave functionr

cp(r) = (R'+ r')-''l'(r),Eq. (7) becomes the eigenvalue problem:(H- v)cp(r) = 0,

where H = Y4(R 2 + r 2)V 2(R 2 + r 2 ). The Hermitian
operator
1
A= (R' + r')-'A'(R' + r') = ml- 3ir +(R'- r')p +2r(rp)],commutes with H, and as A' it satisfies relationshipsof the form given by Eq. (9), i.e., it is a generator of
the 0( 4) group, where
H=R'(A'+L'+'J,).

LA'= A'L = 0.

R'(D- '/,)Ill= viii,

D = _ •j,

The commutator of the Hamiltonian H with the operator A', i.e.,r[H,A']= -2iRH,

H = _ __!_ V'-

2

(12)

If we now transform to the coordinates ~i given by
Eq. (3), and introduce the wave function cl>(~ 1 , ~2, ~ 3 ,~ 4 ) = ( R 2 + r 2 )3/ 2cp ( r ), then by using Eq. (12) we findthat we can write Eq. (11) in the form of the Laplace
equation on the four-dimensional sphere(9)[A,', A/]= ie,jkL,,(11)

L• M..M., + 1.

....

(13)

i',k•tIn this expression Mik is the generator of an infinitesi2v(R'+ r')''(10)

yields zero when it operates on a function satisfying
Eq. (7). The irreducible representations of the 0( 4)group are realized on such functions, and this leads toadditional degeneracy of the energy levels of the problem (H - E)I/J = 0 at the limit of the continuous spectrum. These levels exist only for certain definite
values of v Lsee Eq. (14)j and their degree of degeneracy is n 2 (n = 1, 2, 3, .•. ), which is analogous to theelectron energy levels in a Coulomb field, and contrasts with the degeneracy of ( 2l + 1) for the usual
spherically symmetric potential, where l is theorbital quantum number. Therefore, the symmetrygroup for the fish-eye problem is identical with that

mal rotation on the ~i~k plane of the four-dimensional
space, and 1-D is the angular part of the four-dimensional Laplace operator. The expressions for the operators Mik in terms of the coordinates x, y, z are
given, for example in [ 11 ' 12 l, The eigenfunctions of Eq.
(13) are the four-dimensional spherical harmonics>Itnlm [oJ, and the eigenvalues are degenerate in thequantum numbers l, m:v.=R'(n'- 1/,).

(14)

The Schri:idinger equation for the discrete spectrumof the hydrogen atom in the Fock representation[oJtakes the form
(15)

INTERNAL SYMMETRY OF THE MAXWELL "FISH-EYE" PROBLEM
where K is the integral operator
1
KID(s,, 5,, 5,, 5,) = 2n'fID(s.', sz'. li.s.')d'n'(f.,- s/)' +(£.,- 5,')' +(f.,- s.')' +(£.,- s.')'(16)
In this expression d n' is an area element on the unit
four-dimensional sphere over which the integral isevaluated, <l»(~1, ~2, ~s, ~4) = (R 2 + r 2 )2cpc(r) (we areomitting normalizing factors), ~i can be expressed interms of x, y, z with the aid of Eq. (3), and cpc(r) is
the wave function for the hydrogen-like atom in the
momentum representation in which p is replaced with
4r.
The functions +nlm form a complete set of eigenfunctions for the operators K and D, and comparisonof the eigenvalues
K'P' nzm = n-np nzm,

D'J! nzm = n2'¥ nzm

leads to the result(17)
Therefore, the wave equation for the fish-eye problem
reduces to a differential equation for the four-dimensional spherical functions +nlm, while the Schrodinger
equation for the hydrogen atom in the Fock representation is an integral equation for +nzm. The wave functions for the original problems can be expressed interms of each other'ljl(r) ,= (R'+ r)-'lzrpc(r).

(18)

The wave equation with the potential - 2v/ ( R 2 - R 2 ) 2(with a singularity at r = R) is invariant under inversions in spheres which cut S3 at right-angles. Theinternal symmetry of the problem is described by the
Lorentz group, and so is the continuous spectrum in
the Coulomb problem with which this potential is connected in a similar way to that in which the Maxwell
potential is related to the discrete spectrum.

3. FOCUSING IN THE WAVE PROBLEM AND THE
GREEN FUNCTION
The transformations given by Eq. (8) of the symmetry group of Eq. (7) enable us to find the Green function quite readily. Separating the variables in terms of
the spherical coordinates, we can find the sphericallysymmetric solutions of Eq. (7) with a singularity of theform r- 1 :~j>,(r)=r

which has a singularity of the form I r - r' l- 1 • It isclear that[V' + 4v I (R' + r')']G(r, r') = 4n6(r- r'),i.e., we have found the Green function for Eq. (7). 3>
In[e] the Coulomb Green function in the momentum
representation Fc(p, p', ~) was found in the phaseintegral approximation as the sum over all classicaltrajectories join~ng the points p and p' in the momentum space of the components whose wave behavior isdetermined by the factor expL-iT(p, p', ~)}. In view of
the above connection between the two problems we canreplace the momenta p, p' in the expression forFc(p, p', ~) by the coordinates r, r' and then bymultiplying it by ( R 2 + r 2 312 ( R 2 + r' 2 312 and using
Eqs. (4) and (6) we obtain the quasiclassical Greenfunction for the Maxwell potential. It is interestingthat this differs from the exact Green function given by
Eq. (20) only by the fact that 0! = .J1 + 4v/R 2 has been
replaced by 2 fV!R.
\t,... l/~- \\ ···.;
~-.

r

r

4. ~2l~POTENTIALS AND ADDITIONAL
DEGENERACY
The fish-eye problem can be generalized as follows.
We shall require that all rays leaving a given point on
the sphere S3 and lying in a certain plane passing
through the source and the center of symmetry willpass through some other point on S3 • Let rr/ iJ. be theangle which the two points subtend at the center.
Therefore, rays lying in a given meridional plane will
be focused, but the "foci" for different planes will lie
on a circle. The angle of deflection is given by Eq. (17)in[ 1 l for b = 1, a= iJ.- 1 - 1, and for r < R the refractive index is found to be a special case of Eq. (18) in[ 11 :n(r) = n'~ [ ch (f!ln ~)] _,= ~ (r/R/:(R/r)" ·(21)

If we can extend the validity of Eq. (21) to all space by

2rP

which has singularities of the form I r - r' l- 1 andI r + r'R 2/r' 2 r\ and describes a source placed at the
point r' and its point image (sink) at the point-r'R 2/r' 2 • The existence of this solution means that theMaxwell potential will ensure perfect focusing in awave sense as well, i.e., when diffraction is rigorously
taken into account.
Applying the transformation given by Eq. (8) to
1/! 2 ( r ), we find that
G(r,r') =Re[r(r,r')],(20)

When iJ. = 1 this result becomes identical with Eq. (1).

l'R'+r'(_1 R )/ isin na ,rR exp tatan --;:'(R' + r'(_ R )~j>,(r)=Re[~j>,(r))=---sin atan 1 - ·1085

j sin-,
na2

where a = .J1 + 4v/R 2 • If we apply the inversion given
by Eq. (8) to I/J 1(r), substitute r' = -bR 2/b 2 , and multiply the result by (R 2 + r' 2 112 , we obtain the following
solution of Eq. (7), which is symmetric in r and r':

r

1
1(R' + r') (R' + r'')'I•r(r, r') = tsin(na/2) lr- r'l [ (R' + r') (R' + r'')-lr- r'I'R' ].
[(R' + r') (R' + r'')-lr- r'I'R']''•}
X exp {1a arctg
I,(19)
r-r'IR

analogy with what has been done for Eq. (1 ), we obtain
the refractive index first found by Lenz[ 1sl (seealso[ 4 14 1), or the mechanical problem of the motion ofa zero-energy particle in the potential
2v(22)
U.(r) = - r'R'[ (r/R)"+ (R/r)"]' ·Let us introduce the polar coordinates r, ~ in theplane of a particular trajectory. The equation of the
trajectory is(r/R)" = B sin f!(1'r--!to) +l'B'sin'f!(-lt- tt,) + 1,

(23)3) The expression for the Green function is valid for either sign of v.

Yu. N. DEMKOV and V. N. OSTROVSKII

1086

where the constants J 0 and B are determined from theinitial conditions. If iJ. is a rational number, i.e., if it
can be written in the form of the simple fraction
iJ. = 11 1 /11 2, where 11 1 and 11 2 are integers, theri thefunction r(J) in Eq. (23} is periodic and the trajectorycloses after 11 2 revolutions around the center. In the
opposite case, the trajectory is open and fills densely
a plane region. All the rays leaving the point r are
focused after 11 2 /2 revolutions: 1) at the point-rR 2/r 2 if 11 1 and 11 2 are odd, 2) at the point rR 2/r 2 if
11 1 is odd and 11 2 even, and 3) at the point -r if 111 iseven and 112 odd.
The wave equation with the Lenz potential/2V' + U.(r)]¢(r) =0

(24)

[- 1

where v = VN and

VN=R'!l'(N+ 2~ )(N+ 2~ -1),

(25}

has the following regular solutions which decrease at
infinity:-<"+'Jf'"c<"+'>'••+'ky (tt q>)( ) - ( r )'[( r'i'nlmr -RR )'"+1]n'(R'"-r")R'• + r"

lm

'

(26}
where nr is the radial quantum number (nr = 0, 1,
2, .•• ), n = nr + Z + 1 is the principal quantum number,
N = n + (JJ.- 1 - 1}Z, and c/1 are the Gegenbauer polynomials. Therefore, the values of v for which energy
levels exist at the limit of the continuous spectrum
depend on the quantum numbers n and Z, but only inthe form of the combination n + (JJ.- 1 - 1)Z. This leads
to additional degeneracy of the zero-energy levels if
11 is a rational number. The same reason (dependence
of energy on only the linear combination of quantum
numbers) is responsible for the degeneracy of thelevels of the two-dimensional anisotropic harmonic
oscillator. The possible degrees of degeneracy are
equal to the squares of integers. If we adopt the definition of the symmetry group of a quantum-mechanicalsystem given in[ 151 , the Sturm problems for all the Lenz
potentials with rational I.L have 0( 4) symmetry. The
infinitesimal operators of the group can be constructedby analogy with the case of the anisotropic oscillator. [15 • 161 but the finite transformations of the groupare unknown. 4 > We note that, in the case of two dimensions, the solutions of the wave equation for the Lenz
potential can be obtained from the solutions for the
Maxwell potential with the aid of the conformal transformation used in£ 41 for the corresponding classicalproblem.
5. CONCLUSION
The connection between the closed nature of the
trajectories in classical mechanics and the additional
degeneracy in the corresponding quantum-mechanicalproblem is well known. It is also known that there are
only two central fields, namely, the Coulomb field andthe harmonic field for which the classical trajectories
are closed for arbitrary energy E (this occurs onlyfor E < 0 in the Coulomb field). We have considered
4 >we note that Eq. (24) is invariant under the transformation defined by Eq. (8) when b = 0 (inversion in the sphere S3 ).

'here a class of problems in which the trajectories areclosed only for the fixed energy value E = 0, but withthe coefficient of the potential energy were allowed totake arbitrary values. We could then choose a potential
UI.L ( r) such that the ratio of the periods of the radial
and angular motion will be constant for all trajectories(when E = 0) and equal to any number given in advance.
In the corresponding quantum-mechanical problem wehave the additional degeneracy for E = 0, so that asthe coefficient v in the potential increases, and thepotential well becomes deeper, the new bound states
appear in groups with the same value of the linear
combination of the quantum numbers nr and l, and thedegeneracy is removed as the energy of these states
decreases and departs from the value E = 0. In spite of
the fact that we have confined our attention to E = 0,
the fact that the coefficient in the potential has been
arbitrary enables us to formulate the self-adjoint
problem and obtain a complete set of orthogonal functions for each potential.
In the simplest potential of this kind, i.e., the fisheye potential, the ratio of the radial and angularperiods of the particle is unity, i.e., it is the same asin the case of motion in a Coulomb field. Accordingly,
in the quantum case, we have the same degeneracy,i.e., the energy depends only on the linear combination
nr + Z. Hence, it follows that the symmetry group forthe two problems should be the same, and for the fish
eye the existence of the group of four-dimensional rotations is even more useful than for the Coulomb problem because the stereographic projection in the coordinate space transforms the Schrodinger equation
into the differential equation for the spherical functions,whereas in the Coulomb field we obtain the integral
equation for the spherical functions after stereographic
projection in momentum space.
The fact that the wave equation for the fish-eye
problem can be written as a wave equation on the surface of a four-dimensional sphere provides a particularly useful and unique property of this problem, i.e.,
the fact that the perfect focusing occurs not only in thegeometric but also in the wave approximation. In fact,waves leaving the source located on one pole of thesphere are obviously focused at the opposite pole.
The analysis given here shows that the focusing
properties of the potential are quite general and areclosely related to its internal symmetry. Analysis of
this connection in a general form would be an interesting theoretical problem.
1 Yu. N. Demkov, v. N. Ostrovskii, and N. B.
Berezina, Zh. Eksp. Teor. Fiz. 60, 1604 (1971) lSov.
Phys.-JETP 33, No. 5 (1971)].
2 J. C. Maxwell, The Scientific Papers, New York,
1952, p. 74.
3 C. Caratheodory, Sitzungsberichte derBayerischen Akad. Wiss., Math.-Naturwiss., Abt. No.
1 (1926); Geometrische Optik, Berlin, 1937.
4 R. K. Luneberg, Mathematical Theory of Optics,
Berkeley, 1964.
5 A. Sommerfeld, Mechanics, New York, 1952.
6 M. C. Gutzwiller, J. Math. Phys. 8, 1979 (1967}.
7 G. Gyorgyi, Nuovo Cimento 53A, 717 (1968}.

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INTERNAL SYMMETRY OF THE MAXWELL "FISH-EYE" PROBLEM
8 P. M. Morse and H. Feshbach, Methods of Theoretical Physics, 1953.
9 V. A. Fock, z. Phys. 98, 145 (1935).
10 M, Rotenberg, Ann. Phys. 19, 262 (1962).
11 Yu. A. Dobronravov, Vestnik LGU, No. 10, 5 (1957).
12 M. Bander and C. Itzykson, Rev. Mod. Phys. 38,
330 (1966).
13 W. Lenz, collection: Problems der modernen

Physik, P. Debye, Leipzig, 1928.
14 R. Stettler, Optik 12, 530 (1955).
15 Yu. N. Demkov, Zh. Eksp. Teor. Fiz. 44, 2007
(1963) (Sov. Phys.-JETP 17, 1349 (1963)j.
16 L. A. ll'kaeva, Vestnik LGU, No. 22, 56 (1963).
Translated by S. Chomet
2181087