Non-Bertrandian Systems

Up until about 1963 the theory of accidental degeneracy was rather much as we have outlined it. Of all the textbook systems which were commonly studied, the spherical top, isotropic harmonic oscillator, and hydrogen atom showed far more degeneracy in their spectra than «as required by their readily perceived spherical symmetry. However, it was possible to show that there was in reality additional symmetry present, adequate in each case to account for the observed degeneracy. For the spherical top, it was primarily a matter of understanding clearly that the coordinates were the angular orientation of the top and not he three Cartesian coordinates, and thus not an accidental degeneracy after all. Since the isotropic harmonic oscillator has constant energy surfaces which are pheres in six-dimensional phase space, it is perhaps not surprising that it has a symmetry group consisting of a subgroup of the six-dimensional rotation group which preserves the canonical structure of phase space, which we have seen turns out to be isomorphic to the unitary unimodular group $SU(3)$. Finally, Fock's stereographic mapping interpreted the $O(4)$ symmetry group of the hydrogen atom as another symmetry inherent in phase space, which did not arise from purely geometric transformations in the configuration space. At the same time there were variants on these basic systems of different dimensionality, the anisotropic harmonic oscillator, tops with unequal principal moments of inertia, which could be understood in the same general terms, even if, as in some cases of the anisotropic harmonic oscillator, many ramifications of their symmetry remained obscure.

Additionally, arguments such as the ones we have just cited regarding Bertrand's theorem existed which tended to show that there might not be too many other accidentally degenerate systems, but because they were not stated with mathematical precision, and because there were so many counterexamples when even the most obvious restrictions were violated, the subject could never be quite considered as closed. Among the counterexamples were the problems of motion in non-Euclidean space or in the presence of magnetic fields, which one felt would be similarly restricted when the theory was broadened to include more general types of kinetic energies. Likewise, the requirement of spherical symmetry excluded from the purview of the theory such interesting systems as the anisotropic oscillator, and theoretically important systems such as the cubic square well.

Beyond such tangible uncertainties, one could never be sure that even in those systems for which the degeneracy was supposedly resolved, the day would never come when an even bigger group would be discovered, producing the same pattern of accidental degeneracies, but encompassing constants of the motion which could be important in some as yet unforeseen context. One felt that accidental degeneracy was sufficiently accounted for when he found a hidden symmetry group such that each of its irreducible representations occurred no more than once in the system under consideration, although not all of them need necessarily occur. For example, only the symmetric tensor representations of $SU(3)$ occur in the harmonic oscillator Hamiltonian, and only the representations of dimension $n^2$ of $O(4)$ occur among the hydrogen atom bound states,

When a given representation occurs no more than once in the wave functions of a given Hamiltonian, the mere knowledge of the symmetry group is sufficient to diagonalize completely the Hamiltonian; in fact this is the criterion by which one usually accepts the contention that the hidden symmetry is extensive enough to account for all the degeneracy present. Yet there is no hard and fast rule to judge at what point one has found an adequate amount of symmetry, since it is not feasible to work with the group of ``all operators which commute with the Hamiltonian.'' The extent to which matters have still been left to our judgement has been emphasized by Demkov (22).

We might also remark that such a criterion might be considered too severe. For instance one might find the energy levels of a Morse potential, or even of a mildly perturbed hydrogen atom, and find that the three-dimensional rotation group accounted perfectly for all the degeneracy present, namely the degeneracy in the $z$ component of the angular momentum. Nevertheless, any given representation of the rotation group will occur infinitely often, so that finding wave functions having the symmetry of the symmetry group will by no means determine the eigenfunctions, even though it may simplify the secular equation for the entire system considerably. In this sense we receive considerably more than we bargain for when we determine the hidden symmetry group of the harmonic oscillator or the hydrogen atom.

As a final commentary on the folkloric aspects of accidental degeneracy, we ought to notice that in the harmonic oscillator and the hydrogen atom, the two foremost examples of accidental degeneracy which we have, the constants of the motion are, respectively, a tensor and a pair of vectors. Moreover, the trajectories in the two problems, both ellipses in the bound states, have, respectively, two perihelia and aphelia for the harmonic oscillator, and a single pair for the hydrogen atom. In fact, the existence of an integral number of such maxima and minima in the radial distance plays an important role in the proof of Bertrand's theorem. These observations have sometimes brought forth the wistful thought that there might be a whole hierarchy of tensors, spinors, and such things, each with ``its'' own characteristic potential and whose components might be the generators of a hidden symmetry group for that potential. Thus far this family has never grown beyond its original two members, even though this viewpoint has led to some interesting results concerning the type of potentials which might experience such constants of the motion.

We have been discussing almost exclusively nonrelativistic systems described by Schrödinger's equation. This does not imply that there was no interest in relativistic systems, although there is a far lesser incidence of degeneracy in the physically interesting instances of Dirac's equation. First of all, the relativistic harmonic oscillator does not make much sense on account of its ever-increasing potential, so that only the relativistic hydrogen atom remains, amongst potentials of general interest. But, the relativistic mass change spoils the simple closed elliptic orbits of the nonrelativistic theory, and with them the degeneracy in the principal quantum number of the Dirac equation's eigenvalues. A twofold degeneracy remains, due more to spin effects than to the Coulomb potential, beyond the azimuthal degeneracy to be expected from the spherical symmetry of the Coulomb potential.

Whatever the state of accidental degeneracy, or its theoretical or aesthetic importance, it found an important application around 1958 in the work of Elliott (56) and since that time has permeated the literature of nuclear physics. Of the constants of the motion of the harmonic oscillator, which generated the unitary unimodular group of symmetries $SU(3)$, there were two types which had a long-standing physical significance. Namely, these were the components of angular momentum $L_{ij} = q_ip_j-q_jp_i$ and the energy differences between pairs of coordinates, $D_{ij} = \frac{1}{2}(p_i^2+q_i^2) - \frac{1}{2}(p_j^2+q_j^2)$. The remaining constants of the motion, $K_{ij} = q_iq_j+p_ip_j$, to a certain extent measure the ``correlation'' between the motion in the $i$th and $j$th coordinates, and one may attribute their existence to the fact that the frequency of a harmonic oscillator is independent of its amplitude. Thus, for example, if several particles moving independently in a harmonic oscillator potential are all clustered at the origin in a certain moment, they will all return again to the origin simultaneously, and thus they will seem to exhibit ``collective'' motion, even though they are in reality uncoupled. For this reason the assumption of a harmonic oscillator potential, at least as an approximation to a certain part of the nuclear potential, has found a certain popularity in nuclear physics. Moreover, the fact that the quadrupole interaction has a simple expression in terms of harmonic oscillator constants of the motion has made it a relatively tractable problem to form the proper linear combinations of degenerate harmonic oscillator wave functions, to diagonalize the quadrupole interaction, and to proceed from there with a discussion of nuclear shell theory.

The quantity of literature presently devoted to the nuclear shell model, and its treatment in terms of $SU(3)$ and a wide variety of other Lie groups, makes it impractical for us to make any bibliographical citations, but as a final remark we might note that Moshinsky (57), in 1962, proposed that similar techniques might be found useful in molecular problems, exploiting the $O(4)$ degeneracy of the Coulomb potential. More recently, he has applied the $SU(3)$ techniques directly to such problems.

Another train of thought which started at about the same time as Elliott's was also motivated by the observation of certain regularities in the spacings of nuclear energy levels, and their close similarity to the rotational bands which could be noticed in the spectra of tops of one kind or another. Here the interest was not quite so much in the determination of degeneracies, which had been the traditional role of group theory, but in the actual shape of the spectrum itself. While it was a novel idea to think of Hamiltonians in such terms, the existence of ladder operators for the $z$ component of angular momentum was an example of the way in which a commutation relation between two operators could influence the form of their spectra. Goshen and Lipkin (58) found such pairs of operators among the constants of the motion of the harmonic oscillator, and showed that Hamiltonians which showed rotational bands could be formed by using such constants.

The discovery of a practical application for the theory of accidental degeneracy and the emergence of the idea that operators might determine the shape as well as the degeneracy of the spectrum of a Hamiltonian naturally aroused interest in obtaining a more extensive understanding of accidental degeneracy, especially with the broader view of subsuming the entire spectrum within one irreducible representation of a sufficiently large group. Such a large group would not necessarily be a symmetry group, for as Goshen and Lipkin had shown, considerable information could also be gleaned from operators which did not necessarily commute with the Hamiltonian, provided that the two operators obeyed some other suitable algebraic relationship.

Before we analyze accidental degeneracy more critically, and outline the most recent developments, it is worthwhile to investigate some further problems which have been important in one respect or another, even though the central issue has been neither symmetry nor degeneracy. Such has often been the case when one dealt with unbounded systems, for had there been a symmetry group its irreducible unitary representations would generally have been infinite dimensional, so that such subtleties as the dimensionalities of representations would have gone unnoticed. To start with, since they are exempt from Bertrand's theorem, problems involving the motion of a charged particle in a magnetic field may exhibit a considerable symmetry and accidental degeneracy.