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.
