Phy5646/Group3RelativisticProb

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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:

.