Next: 1.3 Hydrogenic Solutions of Dirac's Equation Up: 1 The Dirac Equation Previous: 1.1 Klein-Gordon Equation

1.2 Dirac's Free Particle
Equation

In order for a wave equation to satisfy the special relativistic requirement
of Lorentz covariance, derivatives in space and time must all appear in the same
order. The K-G equation illustrates that an expression which satisfies this
condition but is non-linear in the space and time derivatives gives rise to
anomalous results. Dirac set out to find an equation which was first order in
space and time derivatives. The result of his efforts, the Dirac equation, is
difficult to motivate and far more complicated that the non-relativistic analog.
However, Dirac's wave equation for a single particle satisfies all the
requirements of special relativity and quantum mechanics, and is able to predict
the properties of one particle systems with remarkable accuracy.

Dirac's equation for an electron in field-free space is given by

where

and is simply the three component momentum. In order to determine the
nature of the three components of and , it is useful to compare the modified equation

to the Klein-Gordon equation. Equations (
1.3) and (
1.6) are equivalent if we enforce the conditions

where represents for i=0 and ,, and for i= 1-3, respectively. In order for this set of four objects to
fulfill these anti-commutation relations, each must be, minimally, four-dimensional.

One set of matrices which obey similar anti-commutation relations are the
Pauli spin matrices: . These are 22 matrices, however, and there are only three of them, so they are not
useful in their usual form. The anti-commutation conditions given in 1.7
are satisfied if the 's are defined as

where is a two by two identity matrix and are the Pauli spin matrices. Though these four by four matrices do not
represent the only set of matrices which satisfy the anti-commutation relations,
any set which does may only differ by a similarity transform from this set.

Because the Dirac equation contains operators represented by four dimensional
matrices, the solutions {} must be represented by a four component vector

The interpretation of the components of this four-vector is not readily
evident. Expressing the Dirac equation in full matrix form provides some
elucidation

The non-relativistic electronic wave-function has two components
corresponding to the and components of spin angular momentum. In the non-relativistic limit,
approaches mc, and the terms which couple with drop out. What remains are four eigenvector equations, with
approximate eigenvalues of for and , and for and . and , then, may be interpreted as the and components of positive energy, electron-like solutions, but the
solutions which are dominated by and do not posses a readily evident interpretation. The existence of these
negative energy solutions troubled Dirac. He felt that it was a ``great
blemish'' on his theory that these apparently unphysical solutions remained.
Over the next few years, however, several experimental investigations, most
notably those of Carl Anderson[1],
gave evidence for the existence of positrons, particles with the mass of an
electron but a positive charge. Experimental confirmation of the existence of
these negative energy solutions was a great triumph for Dirac's theory.

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