Dirac Particle at Rest

We search for a solution to the Dirac equation for a particle at rest. At rest, a particle has an infinitely large de Broglie wavelength and the wave function is uniform over all space ($\hat{p}\psi=0$). Therefore the Dirac equation reduces to

$$i\hbar\frac{\partial\psi}{\partial t} = \beta mc^2\psi .$$ (5.17)

For our representation of $\beta$ (equation 5.10), the solutions are $\psi(\vec{p}=0$):

$$\psi^1(0) = e^{-(imc^2/\hbar)t} \left( \begin{array}{c} 1 \\... ...\left( \begin{array}{c} 0 \\ 1 \\ 0 \\ 0\end{array} \right),$$ (5.18)

$$\psi^3(0) = e^{+(imc^2/\hbar)t} \left( \begin{array}{c} 0 \\... ...left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 1\end{array} \right) .$$ (5.19)

$\psi^1$ and $\psi^2$ are positive-energy solutions, while $\psi^3$ and $\psi^4$ are negative-energy solutions.