Next: 2 Dirac-Hartree-Fock Method Up: 1 The Dirac Equation Previous: 1.3 Hydrogenic Solutions of Dirac's Equation

1.4 Electron Spin Angular Momentum and
Magnetic Moment

One of the triumphs of the Dirac equation was its explicit connection to
electron spin. The Dirac equations does not predict electron spin as a
relativistic property, observable only under relativistic conditions. Reduction
of the Dirac equation for an electron in a magnetic field to its
non-relativistic limit yields the Schrödinger equation with a correction term
which takes account of the interaction of the electron's intrinsic magnetic
moment with the magnetic field. Using the definition of the Dirac Hamiltonian,
, given in (
1.16), we may arrive at the equation

Due to the anti-commutation properties of the components of , we may expand the right hand side to reveal that

If we define our matrices as products of a matrix and the matrices where

so that , then we may express as

where we have used the identity = I. This may be rearranged to the form

via sundry vector identities. The cross-product of with itself does not drop out. This is because the components,
, of the vector are not simple scalars, but rather the sum of a differential
operator, p, and a scalar, A, so the cross terms of the cross products do not vanish, making the
cross-product of with itself

Returning to our original equation 1.47,
we may make this substitution to arrive at

In the non-relativistic limit this gives us a Hamiltonian of the form

The final term indicates that a term which corresponds to a body with a
magnetic moment of - in a magnetic field B must be added to give the proper energy.
This prediction of electron spin as a property which should exist in the
non-relativistic limit provides strong evidence that Dirac's equation provides a
more complete physical picture than the wave equation of Schrödinger, where
electron spin must be treated in an ad hoc manner.

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