Symmetry of the Hydrogen Atom

Fock's paper of 1935, Zür Theorie des Wasserstoffatoms (2), was something of a landmark in this respect, wherein it was shown that just such an explanation could be given for the degeneracies of the hydrogen atom. As a central force problem, the hydrogen atom possesses spherical symmetry. Nevertheless spherical symmetry is only adequate to account for degeneracy in the magnetic quantum number $m$, while the energy of the hydrogenic levels depends only on the principal quantum number $n$, and is independent of the value of the quantum number of the total angular momentum, which may range from 0 to $n-1$. The result is that there is an $n^2$-fold degeneracy for the principal quantum number $n$ rather than a $(2\ell + 1)$-fold degeneracy for the angular momentum quantum number $\ell$.

Such a quantum mechanical degeneracy was reminiscent of a well-known classical degeneracy in the corresponding problem of planetary motion, for it was recognized that the energy of the orbit in Keplerian motion depended only on the semimajor axis of the trajectory, which was always a planar conic section, and not upon its eccentricity. The planarity of the orbit was an established consequence of the conservation of angular momentum, and therefore due to the spherical symmetry of the inverse square Newtonian attraction. However, the orbits of the Kepler problem are almost unique among all the central force problems in that the bounded orbits are simple closed curves, for it seems that only the orbits of the harmonic oscillator share this property. In general such orbits are space filling, a result which was demonstrated by Bertrand almost a century ago. Thus even in classical mechanics the inverse-square force law manifests some exceptional characteristics.

The Hamiltonian for the hydrogen atom is

$${\cal H} = \mbox{$\frac{1}{2}$} p^2 + V(r),$$

where

$$V(r) = -1/r.$$

Fock's demonstration depended upon writing its Schrödinger equation in a momentum representation, as an integral rather than a differential equation

$$\mbox{$\frac{1}{2}$} p^2 \psi(p) - \frac{1}{2\pi^2} \int \frac{\psi(p')(dp')}{\vert p-p'\vert^2} = E\psi(p).$$

In this form, the kernel can be recognized as the Jacobian determinant for a stereographic projection from the surface of a four-dimensional sphere to three dimensions, which in turn suggests writing the Schrödinger equation in terms of angular variables on the hyperspherical surface, all of which finally results in an integral equation which may be recognized as the Poisson kernel for a hyperspherical surface harmonic in the degererate case in which the field point has fallen onto the surface.

By thus placing the hydrogen atom wave functions clearly in evidence as hyperspherical surface harmonics, no doubt is left that the appropriate symmetry group of the hydrogen atom is the four-dimensional rotation group, and not merely the three-dimensional rotation group of the central forces. Strictly speaking one ought to distinguish three cases, according to whether the energy is positive, zero, or negative. The radius of the hypersphere from which the projection is made depends upon the reciprocal of the square root of the negative of the energy, so that only the bound states correspond to an actual hypersphere. In the other cases one deals with either a hyperplane, or the surface of a hyperboloid, so that in those cases the appropriate symmetry group is either an Euclidean group or a Lorentz group.

Immediately after the publication of Fock's paper, Bargmann (3), showed that the generators of the infinitesimal rotations of Fock's hypersphere were nothing other than the components of the angular momentum, together with the components of the Runge vector, when written in terms of the Cartesian components in ordinary three-dimensional space, and their associated conjugate momenta. The components of the angular momentum vector generate the rotations corresponding to the spherical symmetry of the Coulomb potential, so that the additional symmetry present, together with its degeneracy, is seen to be a consequence of the constancy of the Runge vector.

In point of fact, the Runge vector,

$$\begin{eqnarray*} {\bf A} & = & {\bf L} \times {\bf p} + {\bf\hat r}, \end{eqnarray*}$$

where ${\bf\hat r} = {\bf r}/r$ in a unit vector in the radial direction, had been known, under various names, as a constant of Keplerian motion nearly from the time of Newton's original formulation of the law of universal gravitation and the description of the motion of heavenly bodies in terms of the calculus. Perhaps we should more aptly say, it has been known since the time of the origin of such concepts as the vector calculus, which came somewhat after the invention of the calculus itself, by perhaps half a century. The Runge vector is a vector pointing to the perihelion of the orbit, whose magnitude is the eccentricity of the orbit. It is therefore an analytic token of the fact that the orbits for the inverse square law do not process; for other force laws the orbit is typically some sort of rosette.

The earliest published reference to a vector such as the Runge vector which we have been able to find, and which was obviously the precursor of Runge's popular tract on vector analysis, is an article of Hamilton (4) of 1847, communicating a result presented before the Royal Irish Academy in 1845, in which quaternion notation was used to solve the equations of Keplerian motion, and the Runge vector is introduced as a quaternion with zero time derivative. In reality, of course, the fact that we are dealing with an elliptical orbit whose semimajor axis is fixed and which passes through the attracting center which is located at the focus goes back to the observations of Kepler. Nevertheless, we can only begin to touch upon such quantitative aspects as the vector transformation rules or the independence of the orbital energy from the eccentricity, after such mathematical concepts have been appropriately formulated.

Implicit in Hamilton's work are other aspects of the contemporary theory of the symmetry of the hydrogen atom. For example, he formulated the ``law of the circular hodograph'' which states that the hodograph of the Keplerian motion, uniquely among all force laws, is circular. By hodograph is meant the figure resulting from plotting all the velocity vectors of the motion from a common origin. The radius of the hodograph depends on the absolute value of angular momentum, while its plane depends on the direction of the angular momentum vector. Thus if one can assemble all the circles belonging to a common energy into a hyperspherical surface by lifting each one into a fourth dimension, one has something of Fock's momentum space representation.

Vectorial methods gradually came to replace the quaternionic analysis introduced by Hamilton, and set forth in great detail in the two editions of his treatise. Thus we find a solution of the Kepler problem in Gibb's and Wilson's book (5) on vector analysis in which the ``Runge'' vector plays a prominent role, and finally in the widely used monograph on vector analysis of Runge (6), which appears to be the source of the inspiration for its modern usage, even though as we have seen, this was by no means the earliest at which such a vector was known. Nevertheless its existence and convenience in the derivation of the equations of planetary motion were commonly enough known that Pauli (7) was able to make immediate use of it in 1926 in operator form, along with the operator corresponding to the angular momentum, to treat the hydrogen atom by means of Heisenberg's matrix mechanics. He drew in turn upon Lenz' (8) use of the Runge vector in 1924, in conjunction with the angular momentum vector, to describe the hydrogenic orbits according to the old quantum mechanics. Lenz had done so as a prelude to analyzing the perturbation in the motion and spectrum caused by superimposed uniform electric or magnetic fields. Indeed, if we are to believe historical testimony concerning that era (9), the elegance of Pauli's solution was a critical factor in securing the acceptance of matrix mechanics, with its operational methods.

Klein, no doubt remembering his earlier treatment of the spherical top, had been able to comment in 1933 that the components of the angular momentum and the Runge vector together satisfied the commutation rules for the generators of the four-dimensional rotation group, as Hulthén (10), reported in presenting a simplified version of Pauli's derivation. Podolsky and Pauling (11) had exhibited the momentum space wave functions in 1929, and in 1932 Hylleraas (12) had obtained the differential equation for the hydrogenic wave functions in momentum space. So, it would seem that the knowledge of the necessary constants of the motion and their commutation rules was current in the time of Fock's paper. Nevertheless, the integral rather than the differential formulation of the momentum representation and the subsequent introduction of the stereographic projection in momentum space were the essential ingredients in giving a simple geometric interpretation of the true symmetry of the problem. Once this was done, Bargmann immediately made the connection to group theory by relating the symmetries to the transformations generated by the constants of the motion.

Since constants of the motion for the hydrogen atom and the formalities of their commutation rules were known when Fock's paper was written, it is evident that his lucid geometric interpretation of their corresponding symmetries was indispensable in promoting the concept of a ``hidden'' symmetry, if for no other reason than the fact that it was the lack of such a concrete picture which originally motivated the adjective ``accidental'' for the prevailing degeneracies. His technique of stereographic projection invited the scrutiny of other potentials to see whether other degeneracies could receive a similar explanation. In the meantime Laporte and Rainich (13) were investigating a problem in differential geometry which had arisen from a modified electrodynamics proposed by Born, which led to a slightly different concept of minimal hypersurface than the customary one. The study of such surfaces showed that they possessed a type of symmetry in a space of lower dimensionality which could be induced by spherical symmetry in a space of higher dimension if a stereographic projection were made between the two. Laporte (14) showed that these precepts led just precisely to the hydrogenic symmetry which Fock had discovered.

One conclusion to be drawn from the work of Bargmann and Fock was that if it was possible to find a collection of classical constants of the motion whose commutation rules with respect to the Poisson bracket yielded a recognizable Lie algebra, one might hope to find quantum mechanical operators whose commutator brackets might be used for the same purpose. Such a procedure would not be entirely unambiguous, since in the case of the hydrogen atom the commutation rules already depended upon the energy, and it was seen that manifolds of different energy could have different symmetry groups. In this case the Lorentz group, an Euclidean group, or the four-dimensional rotation group arose according to the value of the energy. However, if it should happen that the commutation rules involved some other quantity than the energy, it might be difficult to consider the commutation rules as defining a Lie algebra. Since the energy is a constant of the motion, too serious a problem does not arise when it appears in the commutation rules, since it can always be replaced by its value, classically, and its eigenvalues, quantum mechanically. Another, more serious, problem lies with the correspondence betwesn a classical function of the coordinates and momenta and a quantum mechanical operator, since the quantum operators corresponding to coordinates and momenta do not commute. In the case of the Runge vector, it was possible to find a suitably symmetrized form of the operator by inspection, but one could hardly hope that more complicated constants would succumb to the same improvisation.