Phy5646/Group3RelativisticProb

Free Relativistic Particle

For the Dirac Equation, find plane wave solutions for a free particle.

$i\hbar {\frac {\partial }{\partial t}}\psi =-i\hbar c{\vec {\alpha }}\cdot \nabla \psi +\mathrm {B} mc^{2}\psi$

So, we seek solutions of the form

$\psi ({\vec {r}},t)=Aue^{\frac {i({\vec {p}}\cdot {\vec {r}}-Et)}{\hbar }}$

which is an eigenfunction of both the position and momentum operators. Note that $A$ is a constant, and $u$ is a four-component spinor, independent of the position of the particle.

Putting this general $\psi$ into the Dirac Equation gives a matrix equation:

$(c{\vec {\alpha }}\cdot {\vec {p}}+\mathrm {B} mc^{2})u=Eu$

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) }