Two Coulomb Centers

As we have seen, there is considerable opportunity for the occurrence of accidental degeneracy and hidden symmetry in problems involving motion in non-Euclidean spaces, and velocity-dependent forces. Since the considerations of Bertrand's theorem rule out power law forces other than those of the harmonic oscillator and Coulomb forces, one has to search among the noncentral forces to find further instances of accidental degeneracy. We have already remarked that the presence of magnetic forces, such as in the presence of a monopole or in a constant magnetic field, can originate degeneracies in some instances. However, in treating purely electrostatic forces, one of the first of these which comes to mind is the problem of motion under the influence of two charged particles, the two-center problem. It is a system which admits separation in prolate ellipsoidal coordinates, but which is not classically degenerate. The form of the solution does not change greatly if the force centers bear magnetic as well as electric charge, provided, as in the case of the magnetic monopole, that the repulsive centrifugal potential proportional to the square of the magnetic charge is also included. In ellipsoidal coordinates, the classical turning points are defined by fourth-degree polynomials so that elliptic functions are required to integrate the equations of motion. The orbit will be in general one which fills the volume of the figure of revolution bounded by two ellipsoids and by two hyperboloids.

With such classical antecedents one would not hold much hope for finding quantum mechanical degeneracy beyond that required by the cylindrical symmetry arising from rotating the system about the line joining the two centers, and in the case of equal charge on the two nuclei, reflection in the plane bisecting this line. Nor do the solutions of Schrödinger's equation which are available indicate accidental degeneracy, with the exception that at certain internuclear distances and with certain charges, crossings of energy levels do occur. Such crossings were supposedly forbidden by a theorem of von Neumann and Wigner (101), but it has been known, and recently been shown once again (102) that the existence of the separation constant (103) in spheroidal coordinates produces the vanishing term needed in the Wigner-von Neumann theorem. Alliluev and Matveenko (104) have tried to explain the crossing on the basis of accidental degeneracy, but unfortunately the situation still exhibits a number of complexities.

Part of the reason for this lies, of course, in the fact that the two-center problem is mathematically much more complicated than the simple one-center systems. Classically one finds that he does not obtain a chain of separated equations from the Hamilton-Jacobi equation, as he does for example in the hydrogen atom. In the hydrogen atom the $\varphi$ equation unambiguously determines one constant, the $z$ component of angular momentum, which may be substituted into the theta equation. This equation in turn determines the total angular momentum, which may finally be substituted into the radial equation to extract the energy as the final constant of the motion. However, in the two-center problem although the $z$ component of the angular momentum is readily enough separated, the energy $E$ and the separation constant $\alpha$ appear in both the $\xi$ and the $\eta$ equations ($\xi$ and $\eta$ are the ellipsoidal coordinates). This mixture persists quantum mechanically, so that one obtains two differential equations, each involving two eigenvalues; a much nastier situation than the customary one in which each eigenvalue equation involves only one eigenvalue.

These complications are further compounded by the fact that the $\xi$, $\eta$, and $\varphi$ action integrals are related to the separation constants through complete elliptic integrals so that with the exception of some very special cases, such as one of the charges vanishing, their use as canonical coordinates will involve transcendental functions. Finally one has to contend with the fact that the classical solutions are also expressed in terms of elliptic functions, which makes the formation and discussion of ladder operators that much more complicated.

Historically the first attempt to quantize the two-center problem seems to have been made by Pauli (105), in 1922, using the old quantum mechanics, and it was not entirely successful -- a failure which hastened the decline of the old quantum mechanics. The reason for this probably lies in the very nature of the two-center problem. Classically a tightly bound electron will circulate near one center or the other, but in quantum mechanics that is another matter. If the centers are near together, even though a wave packet is initially constructed with the particle's wave function concentrated at one center, it will eventually and perhaps rather rapidly diffuse into the neighborhood of the other. The eigenfunctions will have nonzero amplitude near both centers, and thus the electronic behavior is classically rather different from the quantum mechanical version.

In any event, wave mechanical calculations have been undertaken since the advent of the new quantum mechanics, and have continued until the present time. They may be grouped into three broad categories: direct attempts to integrate the Schrödinger equation, molecular orbital calculations, and semiclassical computations which are surprisingly simple and accurate.

There are two limiting cases with which comparisons can readily be made: the ``united atom'' in which the centers are very close in comparison to the potentials involved, and the ``separated atom'' in which they are very distant. In the former case, one has small perturbation of a dipole or quadrupole type of an atom with the total charge of the two nuclei, and thus very nearly a single hydrogen atom. In the other limit the system behaves as though it were composed of two atoms which influence each other through a weak Stark effect.

One may hope to interpolate between these two limits, and compare the levels with one symmetry at one limit with similar levels at the other limit. It is at this point that the ``no crossing'' theorem of Wigner and von Neumann enters. The only difficulty is that crossing, in fact, occurs, and thus there can be accidental degeneracy of certain levels at certain internuclear distances. The possibility for a more exact investigation of the entire degeneracy situation of the two-center problem still remains open.

In fact, one might remark somewhat on the philosophy of the two-center problem as it most often occurs in the literature. The interest has been not so much to find the complete set of energy levels for a nuclear pair at a fixed internuclear distance as it has been to find the energy of the ground state as a function of the nuclear separation. In this way one may hope to determine the binding energy of a diatomic molecule, at least one of such simple structure, and related information.

A limiting case of the two-center problem is the electric dipole, which has received attention in recent times due to the curiosity which exists as to whether or not it has any bound states. Here the problem is complicated by the fact that the potential of the dipole is a singular potential, meaning that it is of the form $r^{-k}$, for $k \ge 2$. This radial term is also modified by an angular factor in the dipole problem. Classically, the singular potentials possess divergent radial action integrals, so that a complete set of action-angle variables cannot be found! The precise case of $k = 2$ has the unusual feature that there is a continuum of eigenvalues, even for the bound states. Case (106) has shown how one can still quantize such singular potentials, but additional information beyond the Hamiltonian -- such as a specification of phase at the origin -- is required. Classically the situation is rather similar, as it would not seem that Newton's equations alone are adequate to account for the particle's passage through the origin.

Singular potentials, and particularly the centrifugal potential, make an appearance in several situations. The classical motion under the centrifugal potential is a spiral, whose extent depends upon the energy of the particle. If the particle is unbound, it will spiral away to infinity, while if it is bound, it will spiral in to the origin and out again, but it spirals in at a constant angle to the radius vector and its precise position of emergence cannot be foreseen from Newton's laws. Some orbits are circular, but these are unstable against the slightest perturbation. An interesting feature of the Hamiltonian is its invariance under dilations. Any orbit may be dilated to obtain another orbit, although the time scale must simultaneously undergo a reciprocal adjustment.

This dilational symmetry manifests itself in the Schrödinger equation, in that a wave function may be dilated by any amount to obtain another wave function. Thus even though the wave functions are normalizable, one does not obtain a quantization of energy unless some supplementary condition is inserted.

Sometimes it is convenient to construct artificial problems by adding to a known potential some centrifugal terms, and sometimes such potentials arise naturally. We have seen the convenience of adding such a potential to the monopole problem, and later on we shall describe the results of making such a modification to the harmonic oscillator and Kepler potentials. In the case of the Dirac equation, the relativistic precession owes its existence to a naturally occurring term of this sort in the squared Dirac equation. It may happen that the centrifugal potential overwhelms the Coulomb potential in problems where they both are found together. The Coulomb potential predominates at long ranges, and so if it is attractive, there will result a bound region even for a repulsive centrifugal potential, which in turn predominates at short ranges. However, if the repulsive centrifugal potential is too large, the bound states may be lost, and if it is too attractive and counterbalances the natural centrifugal forces, the continuum we have mentioned may arise. This would occur for example in the Dirac equation for nuclei with charges greater than 137, although this is considerably beyond the limit of the heaviest nucleus now known. There has been some mention of such a possibility in the literature.

The unstable nature of the circular orbits in a centrifugal potential is matched by a similar situation in the field of a point electric dipole, wherein there exist semicircular orbits. Again, the slightest perturbations will cause these orbits to pass over into orbits which fall into the origin or recede to infinity. This point was responsible for some interesting discussions in the days of the old wave mechanics, when the mechanism of quantization of the radiation from atoms was still not well understood. Thompson (107) had proposed that atoms, which have a generally attractive central potential, but which could certainly be modified by an additional dipole if the charge were not symmetrically located at the nucleus, might possess electronic orbits in which the electron oscillated about a position of equilibrium off to one side of the atom. Such an oscillation depended upon the assumption that the semicircular orbits were stable, and that small displacements from them would result in bounded motion; the point was disputed in a rather interesting series of articles (108a-g) mostly by Higab (l08d-f), in the Philosophical Magazine around 1930.

As for the case of the point dipole, it seems that if the dipole is finite, it is possible to have some bound orbits, which pass between the two nuclei. But, when one arrives at the limit of a pure dipole. if the dipole is not adequately strong, there can be no bound state.

The limiting case in the opposite direction is the Stark effect, in which one nucleus recedes to infinity as its charge grows, resulting in having a uniform field acting on the remaining nucleus. Redmond (109) has shown the connection of the separation constant in parabolic coordinates with the Runge vector, which we might compare with a rather similar relation found by Coulson and Joseph for the two-center problem. In the case of the Stark effect it is curious to note that if the energy is written in terms of the action-angle variables, one still finds a classical twofold degeneracy, which replaces the threefold degeneracy characteristic of the hydrogen atom. Since the threefold degeneracy is responsible for the accidental degeneracy of the spherically symmetric Coulomb potential, this residual twofold degeneracy should still result in accidental degeneracy in the cylindrically symmetric potential of the Stark effect. Bethe and Salpeter's Handbuch article (110) quotes results indicating that this degeneracy persists at least into the first orders of the quantum mechanical Stark effect.

As yet relatively little detailed investigation of the symmetry, possible ladder operators, and so on seem to have been made for the two-center problem or its limiting cases. Some other simple problems which likewise have not been much studied but which might show some interesting aspects are the motion of a charge particle in the field of an electromagnetic wave or a variant of the same theme which is to study the motion of a charged particle in the combination of a uniform magnetic field and an electromagnetic wave. In neither of these cases is either the symmetry or the degeneracy of the motion particularly known, although the orbits are reasonably regular.