We should find four solutions. Lets start with one that gives a

Use the third and fourth components to solve for the coefficients and plug them in for a check of the result.

This will be a solution as long as , not a surprising condition. Adding a normalization factor, the solution is.

This reduces to the spin up positive energy solution for a particle at rest as the momentum goes to zero. We can therefore identify this as that same solution boosted to have momentum . The full solution is.

We again see that for a non-relativistic electron, the last two components are small compared to the first. This solution is that for a

If we make the upper two components those of a **spin down electron**, we get the next solution following the same procedure.

This reduces to the spin down positive energy solution for a particle at rest as the momentum goes to zero. The full solution is.

Now we take a look at the **``negative energy'' spin up solution** in the same way.

This reduces to the spin up ``negative energy'' solution for a particle at rest as the momentum goes to zero. The full solution is

with being a negative number. We will eventually understand the ``negative energy'' solutions in terms of anti-electrons otherwise known as positrons.

Finally, the **``negative energy'', spin down solution** follows the same pattern.

with being a negative number.

We will **normalize the states** so that there is one particle per unit volume.

We have the **four solutions with for a free particle with momentum **.
For solutions 1 and 2,
is a positive number.
For solutions 3 and 4,
is negative.

The spinors are

It is useful to write the plane wave states as a spinor
times an exponential.
Sakurai picks a normalization of the spinor so that
transforms like the fourth component of a vector.
We will follow the same convention.

Are the free particle states still eigenstates of
as were the states of a particle at rest?
In general, they are not.
To have an eigenvalue of +1, a spinor must have zero second and fourth components
and to have an eigenvalue of -1, the first and third components must be zero.
So **boosting our Dirac particle to a frame in which it is moving, mixes up the spin states**.

There is one case for which these are still spin eigenstates.
If the particle's momentum is in the z direction, then we have just the spinors we need to be eigenstates of
.
That is, if we boost along the quantization axis, the spin eigenstates are preserved.
These are called **helicity eigenstates**.
Helicity is the spin component along the direction of the particle.
While it is possible to make definite momentum solutions which are eigenstates of helicity,
it is not possible to make definite momentum states which are eigenstates of spin along some other direction
(except in the trivial case of
as we have shown).

To further understand these solutions, we can compute the conserved probability current. First, compute it in general for a Dirac spinor

Now compute it specifically for a positive energy plane wave, , and a ``negative energy'' plane wave, .

Since is negative for the ``negative energy'' solution, the

Jim Branson 2013-04-22