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 4
4 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.
