Principles of Quantum Mechanics · Problem 20.4

The Dirac Equation — R. Shankar

Physics — Quantum Mechanics · dirac-equation-free-particle

Problem

Find the positive-energy plane-wave solutions of the free-particle Dirac equation and show that they can be written as

\psi = N \begin{pmatrix} \phi \\ \dfrac{\boldsymbol{\sigma}\cdot\mathbf{p}c}{E + mc^2}\phi \end{pmatrix} e^{i(\mathbf{p}\cdot\mathbf{r} - Et)/\hbar}

where

\phi

is a two-component spinor and

N

is a normalization constant.