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Up: 1 The Dirac Equation Previous: 1.2 Dirac's Free Particle Equation

1.3 Hydrogenic Solutions of Dirac's
Equation

In its present form, the Dirac equation cannot even address the hydrogen
atom, since it has no provision for an external potential. If we introduce
scalar and vector potentials, and A, to the momenta of the field free Dirac equation

then the Dirac equation becomes

If the potential terms are not explicitly time dependent, we may make the
rearrangement

The left hand side defines the Dirac Hamiltonian for a single particle in a
time-independent electro-magnetic field. In the absence of a magnetic field the
Dirac Hamiltonian, , may be expressed in full matrix form as

This Hamiltonian is now ready to take on the task of an electron in the field
of a nucleus.

Because of the spherical symmetry of the nuclear potential, the Schrödinger
equation for the hydrogenic atom may be simplified via separation of variables
into a radial equation and two angular equations. The angular solutions are
given by the spherical harmonics, , and the radial solutions by

where are the associated Laguerre polynomials, and n, l, and
are the principal quantum number, orbital angular
momentum quantum number, and the z-projection of l, respectively.

The hydrogenic solutions to the Dirac equations may also be achieved
analytically, though the resulting eigenfunctions cannot be so simply expressed.
Because of the presence of the term, the and operators do not commute with . The components of the operator, however, do commute with , and so the eigenfunctions of the Hamiltonian may be eigenfunctions of
and . and also commute with the Hamiltonian, and so we may still associate a
particular l and s value with each solution, .

It is useful, at this point, to express the four-component wave function as a
pair of two component spinors, and :

Such a separation is dictated by the nature of the 44 Dirac spin matrices, and makes it possible to work with the familiar
two-spinors of non-relativistic theory. Capitalizing on the commutation of with , each spinor may be expressed as the product of a two-component
eigenfunction of which is dependent upon the spin and spatial angular coordinates and a
radial function:

The eigenfunctions of , , may be expressed as sums of direct products of the more familiar
eigenfunctions of and and the appropriate Clebsh-Gordon coefficients:

where are the spherical harmonics and

Next, it is useful to define the operator

which commutes with . The eigenvalue equation associated with is given by

where

Because of the spherical symmetry of the nuclear potential, the
eigensolutions may be further separated on the basis of their response to
spatial coordinate inversion. The parity operator commutes with the full
Hamiltonian, and so the final eigenfunctions, , must also obey the eigenvalue equation

This necessarily dictates one of the following forms for the four-component
eigensolutions:

It is possible, through the use of the identity

and the property

to completely separate out the angular functions from the eigenvalue
equation, and thereby achieve two coupled, complex differential equations for
the radial functions and . The resultant equations are greatly simplified by the substitutions

to yield

The solutions to these differential equations may be determined by first
solving the asymptotic equations in the limit as . These solutions, which are of the form

are then multiplied by an undetermined power series for both components and
the bound state solutions are sought.[2]
The final form of each wave function is uniquely determined by the recursive
relations of the power series and the boundary conditions, but may not be
expressed in a simple general form. The resulting electronic energies are given
by

where is the fine structure constant, and is given by in atomic units.

The four-component nature of the Dirac eigenfucntions gives rise to many
interesting differences when compared to the non relativistic solutions. In
order to better understand these differences, it is useful to inspect some
simple approximations of the lowest energy hydrogenic solutions, for example the
n = 2, j = { 1/2, 3/2} solutions.

These solutions are correct to order Z, as illustrated in Moss's discussion of the hydrogenic solutions to
Dirac's equation[2].
Here, analogous to the non-relativistic case, for each unique combination of the
principal quantum number n and a specific angular momentum quantum number
j, the solutions must be further classified according to the projection
of their total angular momentum on the z-axis, . These solutions are, of course, degenerate in the absence of an
external field. The contributions from the first two components of the
wave-function tend to be much larger than the contributions from the final two
components for electron-like solutions. For this reason the upper two-component
spinor is known as the large component of the wave-function, while the lower
spinor is known as the small component. By orthogonality, the opposite is true
for the positron-like solutions. Since the angular and spin portions of the
wave-functions are normalized, the magnitude of the large and small
contributions is determined entirely by the radial functions, and . Given the asymptotic form of the radial equations (
1.34) and the orbital energy (
1.36), the ratio of their contributions may be expressed as

In general, the ratio will probably be larger than this approximate value in
regions close to the nucleus, where the contribution of is the greatest. The contribution of the small component, then, may be
significant for sufficiently heavy nuclei. Though hydrogenic ions with very
heavy nuclei may not be of great practical interest, this relationship will have
implications for heavy many-electron atoms where the innermost electrons
experience a large portion of the full nuclear charge and hence are closely
related to the analogous hydrogenic systems.

One interesting consequence of the four component wave-function may be
observed in the associated probability density. Scalar wave-functions, , which possess radial and angular nodes will have the
same radial and angular nodes in their probability density, The four-component wave-function, however, gives rise to a probability
density which is the sum of the probability densities associated with the large
and small wave-functions, and , respectively. The radial component of and posses different numbers of nodes and, in general, none of these nodes
will coincide. Similarly, the angular contributions of differ from those of by a single unit of angular momentum. Therefore, although and may possess either radial of angular nodes individually, their sum,
their sum will be node-less. This is illustrated by Figure 1.3,
where the large and small component radial functions for the n = 2,
j = 1/2, k = -1 orbitals, g(r) and f(r), have been plotted, along
with the resultant radial probability density, = g(r) + f(r).

The energy eigenvalues of the hydrogenic solutions to the Schrödinger
equation are only dependent upon n, the principal quantum number, while
the Dirac hydrogenic eigenvalues are dependent on both n and j.
The spectral dependence on j is upheld by experimental observation,
however the degeneracy of eigenvalues for solutions with the same j
values but differing l values is not observed in nature. The breaking of
this degeneracy is known as the Lamb shift[3]
and its origin has been attributed to the difference in the interaction of the
different j eigenfunctions with the vacuum fluctuations predicted by
quantum electrodynamics. The magnitude of the Lamb shift is much smaller than
the splitting introduced by spin orbit coupling and is largest for the n
= 2, j = 1/2 shells of the hydrogenic atom. In this case, the splitting
introduced by the Lamb shift is approximately 10% of the magnitude of the energy
separation of the and eigenfunctions.[2]
In order to understand the source of the Lamb shift, it is necessary to appeal
to quantum electrodynamics, where higher order couplings of the electromagnetic
interactions of the electron and nuclei are taken into account.

Next: 1.4 Electron Spin Angular Momentum and Magnetic Moment
Up: 1 The Dirac Equation Previous: 1.2 Dirac's Free Particle Equation

This document is copyright 1996,
Thu May 29 08:12:02 EDT
1997