The Dirac equation
In this chapter the Dirac equation will be discussed, which is neccessary for
the quantum field theoretical description of fermions, such as electrons.
To start with, the Schrödinger equation will be
revisited and a relativistically invariant formulation, the Klein-Gordon equation
will be introduced. This equation, however, has a number of intrinsic problems,
tied to the fact that it is quadratic in the derivative of the field w.r.t. time.
Historically, these problems led to the Dirac equation, which is linear in this
and the other derivatives.
Independently of that, it will become obvious that the Klein-Gordon equation, being
an equation for scalar fields, does not describe particles with internal degrees
of freedom such as spin. Thus, in the next part, spin will
be briefly discussed.
Only then, the Dirac equation in its full wealth will
be presented and discussed, and finally, various
ways to represent its solutions will be investigated.