Dirac Equation for the Hydrogen Atom

Interesting as the solutions of Dirac's equations may be for plane waves, and even for the motion of a charged particle in a uniform magnetic field, it is in the treatment of the hydrogen atom that the equations show their true utility, and also where they have resisted for the longest time the analysis of their symmetry. This recalcitrance was apparent from the start, for rigorous solutions to the equations for the hydrogen atom were not obtained until after some of the simpler systems had first been considered, and it was verified that the magnetic moment of the electron arose as a natural consequence of this type of wave equation. However, once the hydrogenic equation was analyzed, operational techniques were applied from the very beginning, along with more traditional methods of series solution. Such methods may be found in the early papers of Temple (123), in his small monograph (124) on quantum mechanics, as well as in some contemporary publications of Sauter (125).

On trying to construct a relativistic wave equation for systems in which there are interactions between various particles, the first approach is the usual one, to assume that the particles interact through fields, and that one may therefore to a first approximation treat the motion of a single particle in a fixed field, the most important being the electromagnetic field. The electromagnetic field may be introduced into Hamiltonian mechanics by adding the vector potential to the canonical momentum, but if one is to follow relativistic analogy in its entirety, the electrostatic field ought to be added to the Hamiltonian as well since it represents the fourth component of a vector in Minkowski space. In that way the Dirac equation may be formulated for the relativistic hydrogen atom, choosing a Lorentz frame in which the potential is purely electrostatic.

Although the Dirac equation is a covariant equation with respect to general Lorentz transformations, which means that the transformation rules for vectors are to be used in converting it from one coordinate system to another, when it is applied to a specific example such as the hydrogen atom which has spherical symmetry, one should expect to find an invariance to rotations.

However, for the same reasons that we have described, concerning the plane wave, the fact that we are dealing with a first-order system of equations and not with differential operators formed from group invariants precludes our finding such a symmetry. Here also we find that a rotation of the spatial coordinates must be coupled with a simultaneous mixing of the spin components, with the result that again it is the total angular momentum which is the conserved quantity and which generates a symmetry transformation. The orbital angular momentum by itself is not conserved, which implies the existence of a spin angular momentum in the Coulomb potential as well as for the free particle.

On account of the precession of the hydrogenic orbits in relativistic classical mechanics, one does not really expect to find that the Runge vector is a constant of the motion of the Dirac equation, nor that the accidental degeneracy of the hydrogen atom remains. This loss of degeneracy is in accord with the experimentally observed hydrogenic spectrum, and is confirmed by the solutions of the Dirac equation, which give very close agreement with the results. Perhaps for this reason symmetry considerations have never been pushed very far for the Dirac equation of the hydrogen atom, but there nevertheless remains a very mysterious two-fold degeneracy, above and beyond the degeneracy which one would expect from the spherical symmetry and the conservation of the total angular momentum. This degeneracy seems to be characteristic of the Dirac equation, but not of the Klein-Gordon equations, and is not uniquely confined to the Coulomb potential as had sometimes been thought. We shall see that it depends on the existence of the spin angular momentum, and represents a degeneracy in what might be called the ``helicity'' of the Dirac wave functions.

The Dirac equation for the Coulomb potential has the form

$${\cal H}(m)\Psi = \left({\bf\alpha \cdot p} + \beta m - \frac{\alpha Z}{r} \right) \Psi = - E \Psi.$$

However, the usual method for solving the Dirac equation is to construct the mass annihilation operators

$$\begin{eqnarray*} {\cal O}_+ & = & \rho_3[{\cal H}(m) + E ], \\ {\cal O}_- & = & \rho_3[{\cal H}(-m) + E ], \end{eqnarray*}$$

and to observe that any solution of the Dirac equation must also be a solution of the ``squared'' equation

$${\cal O}_+{\cal O}_- \Psi = 0,$$

which in the present instance takes the form, written in polar coordinates

$$\left[\frac{1}{r^2}\frac{\partial}{\partial r}r^2\frac{\part... ...\alpha Z\rho_1 {\bf\sigma \cdot \hat r}) + K^2\right]\Psi = 0,$$

where $K = \rho_3({\bf\sigma \cdot L} + 1)$.

Here, the role played by two terms is to be noticed. The term $(K^2 - (Z\alpha)^2)$ may be thought of as a term in the square of the angular momentum causing a relativistic precession by slightly splitting the energy of the different states of a given total angular momentum. This term appears in the Klein-Gordon equation, but does not give the experimental amount of fine structure splitting.

That it does not do so is due to the term dependent on

$$\Gamma = \rho_3 K + i \alpha Z \rho_1 {\bf\sigma \cdot \hat r}.$$

In fact, if the equation is rewritten in terms of $\Gamma$, it acquires the very suggestive form

$$\left(\frac{1}{r^2}\frac{\partial}{\partial r}r^2\frac{\part... ...ma(\Gamma-1)}{r^2} - \frac{2k\eta}{r} + k^2\right) \Psi = 0,$$

where $k^2 = E^2 - m^2$, and $\eta = \alpha Z E/k$.

As a first remark, let us say that the operator $K$ occupies a role quite similar to the angular momentum operator, and in fact has eigenvalues which are substantially the eigenvalues of the total angular momentum. It may be demonstrated to be a constant of the motion, whose eigenvalues may be used to classify the quantum states of the wave functions. In fact, the twofold accidental degeneracy of the Coulomb potential arises out of the fact that the energy depends only upon the absolute value of the eigenvalues of $K$, whereas wave functions occur belonging to eigenvalues of $K$ of both signs. For a long time, only ${\bf J}$, ${\bf J}^2$ and $K$ were known as constants of the motion, leaving pen the possibility for speculation as to what other constant of the motion might exist that would be responsible for the additional degeneracy. Such a constant was found in 1950 by Johnson and Lippman (126), with the interesting property that it commuted with the Hamiltonian, but anticommuted with $K$. It therefore simultaneously forced $K$ to have negative pairs of eigenvalues, and made them degenerate in energy.

Johnson and Lippman's constant probably exists for all spherically symmetric potentials, and certainly does so for a plane wave, where it measures the projection of the spin on the momentum of the particle. For the Coulomb potential, it seems to correspond more nearly to the projection of the spin on the Runge vector, but in any event its physical significance seems to be that it determines the relative orientation of the spin and some reference vector in the system.

Several methods exist for solving the squared Dirac equation, of which some noteworthy instances are the methods of Hylleraas (127) and Kolsrud (128). One version of considerable historical antiquity is simply to determine the eigenfunctions of the operator $K$, which correspond to the angular part of the equation. They are not difficult to determine, inasmuch as they are the same spin functions which had arisen earlier in Pauli's theory of the spinning electron. Once the angular part of the equation is separated, there remain some coupled equations for the radial functions, whose resolution by various schemes forms the substance of the various papers mentioned above. One of the more comprehensive recent treatments was made by Martin and Glauber (72°), and later extended by Biedenharn (130).

Biedenharn's proposal was to diagonalize the operator $\Gamma$, which may be accomplished by a Foldy-Wouthuysen transformation applying only to $\Gamma$ and not to the entire Hamiltonian. It was found that the radial wave functions occurring in such a representation were exactly like the nonrelativistic Coulomb wave functions, with the difference that the eigenfunctions of angular momentum which are half-integers or integers in the nonrelativistic treatments are modified by the fine structure. In other words, there resulted a considerably simplified way of obtaining the radial wave functions of the Coulomb problem.

As we have already explained in connection with the description of the magnetic monopole problem, it can often be quite instructive to consider an alternative system which might not be physically realistic, but which nevertheless might have some exceptional property worthy of study, such as an extraordinarily high degree of symmetry. A similar idea occurred to Biedenharn, to modify the Coulomb problem in such a way as to obtain a degenerate Hamiltonian, if possible one which retained the nonrelativistic degeneracy of the hydrogen atom. We have already seen that there exists such a related classical Hamiltonian, which not only has the same degeneracy but the same symmetry group. The modification which suggests itself is simply to replace the operator $\Gamma$ by an operator having desired eigenvalues, such as $K$ itself. However, the framework of the Dirac equation requires this to be done with considerable finesse, for the alternative system which we hope to obtain is to be described by a modified Dirac Hamiltonian, and not by a modification in the squared equation. Thus, a modification which is readily apparent in the squared Hamiltonian will required some ingenuity to take account of the noncommutation of quantum mechanical operators and the functional dependence of the squared Hamiltonian on the original Hamiltonian.

Nevertheless, Biedenharn (131) succeeded in discovering the term which was necessary to add to the Coulomb Hamiltonian,

$${\cal H}_{fs} = \rho_2 \frac{\sigma\cdot\hat{\bf r}}{r} K \... ...left[1+\left(\frac{\alpha Z}{K}\right)^2\right]^{1/2}-1\right)$$

and another Foldy-Wouthuysen transformation which diagonalized the angular momentum term in the resulting squared Hamiltonian. The details were set forth by Biedenharn and Swamy (132), in the paper in which they introduced their ``symmetric'' Hamiltonian.

Sheth (133), in 1968, took the Foldy-Wouthuysen nonrelativistic limit of the symmetric Hamiltonian, to see that it approaches the Hamiltonian we mentioned in the introduction to this section, in which the relativistic precession is canceled by a vector potential whose magnetic field causes a counterprecession. It is worthy of note that there are two relativistic effects which occur in the orbital precession of the hydrogenic orbits. One is, let us say, an average mass deviation which we could attribute to the relativistic mass change a particle moving at the average orbital velocity would experience. It would even be possible to interpret this effect in terms of a modified central nuclear charge rather than in terms of a mass variation of the electron. This is the variation which Sommerfeld called the relativistic correction for circular orbits. Additionally, there is the further mass variation which changes with the particle's position in the orbit, and which leads to the precession. This division of the relativistic effects into two parts affects the nature of the approximations which one might use in calculating the relativistic Coulomb wave functions. Biedenharn and Swamy found that the perturbation series which would result from removing the fine structure interaction from the symmetric Hamiltonian was not quite the same series which was found by Furry, Sommerfeld and Maue (134), Bethe and Maximon (135), and others who were concerned with incorporating relativistic effects into the Coulomb wave functions through series expansions.

The symmetric Hamiltonian shows $O(4)$ degeneracy, and it is apparent that the angular momentum and the Runge vector are constants of the motion generating this degeneracy when an appropriate coordinate system is used. To exhibit these constants in the customary Dirac representation is a relatively cumbersome task, but in any event, Biedenharn and Swamy did exhibit a set of operators generating the $O(4)$ group in their paper.

Even in the symmetric Hamiltonian, the complete degeneracy is not merely the $n^2$ degeneracy of the group $O(4)$, but it is $2n^2$, double the expected amount. Again it is found that there is a twofold $K$ degeneracy. Its presence can be explained by an operator which, like the Johnson-Lippmann operator, is a constant of the motion and anticommutes with $K$. Whereas Biedenharn and Swamy showed that their symmetric Hamiltonian has two vector constants of the motion generating an $O(4)$ symmetry group, the symmetry is not as simple as it appears. Although the total degeneracy is $2n^2$, it arises from the accidental degeneracy of a representation of dimension $n(n + 1)$ of $O(4)$ with one of dimension $(n-1)n$, so that one has not found a degeneracy which could have an interpretation in terms of wave functions defined on a hypersphere.

In the most delicate experiments, it is found that not even the $K$ degeneracy occurs, the very small difference in energy between these levels which occurs being the Lamb shift, which was measured precisely enough to establish the nondegeneracy with certainty in 1948. However, it is due to effects not treated at the level of approximation of the Dirac equation - interaction of the electron, with a quantized radiation field being the essential omission.