To study this further, lets take the simple case of the free particle at rest.
This is just the
case of the the solution above so the energy equation gives
.
The Dirac equation can now be used.

This is a very simple equation, putting conditions on the spinor
.
Lets take the case of positive energy first.

We see that the **positive energy solutions**,
for a free particle at rest, are **described by the upper two component spinor**.
what we have called
.
We are free to choose each component of that spinor independently.
For now, lets assume that the **two components can be used to designate the spin up and spin down** states
according to some quantization axis.
For the ``negative energy solutions'' we have.

We **can describe two spin states for the ``negative energy solutions''**.
Recall that we have demonstrated that the first two components of
are large compared to the other two
for a non-relativistic electron solution and that the first two components,
, can be used as the
two component spinor in the Schrödinger equation (with a normalization factor).
Lets identify the first component as spin up along the z axis and the second as spin down.
(We do still have a choice of quantization axis.)
**Define a 4 by 4 matrix which gives the z component of the spin**.

With this matrix defining the spin, the **third component is the one with spin up** along the z direction for the
``negative energy solutions''.
We could also define 4 by 4 matrices for the x and y components of spin by using cyclic permutations of the above.
So the four normalized solutions for a Dirac particle at rest are.

The **first and third have spin up** while the second and fourth have spin down.
The first and second are positive energy solutions while the
**third and fourth are ``negative energy solutions''**, which we still need to understand.

Jim Branson
2013-04-22