We can show that a free particle solution can be written as a
**constant spinor times the usual free particle exponential**.
Start from the Dirac equation and attempt to develop an equation to show that each component has the free particle
exponential.
We will do this by making a second order differential equation, which turns out to be the Klein-Gordon equation.

The

Note that the momentum operator is clearly still and the energy operator is still .

There is no coupling between the different components in this equation, but,
we will see that (unlike the equation differentiated again)
the **Dirac equation will give us relations between the components of the constant spinor**.
Again, the solution can be written as a constant spinor, which may depend on momentum
, times the exponential.

We should normalize the state if we want to describe one particle per unit volume: . We haven't learned much about what each component represents yet. We also have the plus or minus in the relation to deal with. The solutions for a free particle at rest will tell us more about what the different components mean.