Free Relativistic Particle
For the Dirac Equation, find plane wave solutions for a free particle.
So, we seek solutions of the form
which is an eigenfunction of both the position and momentum operators. Note that
is a constant, and
is a four-component spinor, independent of the position of the particle.
Putting this general
into the Dirac Equation gives a matrix equation:
Now, it becomes convenient to write the four-spinor u using two two-component spinors:
Failed to parse (unknown function "\begin{array}"): {\displaystyle u = \left(\begin{array}c u_a \\ u_b \end{array} \right); u_a = \left(\begin{array}c u_1 \\ u_2 \end{array} \right); u_b = \left(\begin{array}c u_3 \\ u_4 \end{array} \right) }
Our equation is now written:
Which gives that
and
are related by:
Substituting these equations into each other gives:
and
And, solving for the eigenenergies yields (from the two roots):
Where each one occurs twice, once for each component of the two-component spinors.
Now, solving the two-component matrix equations for
and
gives for the four components of the spinor:
Which are clearly normalized by:
.