Advanced Quantum Mechanics Schwabl · Problem 6.4

The Coulomb Potential

Physics — Quantum Mechanics · quadratic-dirac-equation-electromagnetic-field

Problem

Derive the quadratic form of the Dirac equation

\left[\left(\mathrm{i}\hbar\partial - \frac{e}{c}A\right)^2 - \frac{\mathrm{i}\hbar e}{c}(\boldsymbol{\alpha}\boldsymbol{E} + \mathrm{i}\boldsymbol{\Sigma}\boldsymbol{B}) - m^2c^2\right]\psi = 0

for the case of external electromagnetic fields. Write the result using the electromagnetic field tensor

F_{\mu\nu} = A_{\mu,\nu} - A_{\nu,\mu}

, and also in a form explicitly dependent on

\boldsymbol{E}

and

\boldsymbol{B}

. Hint: Multiply the Dirac equation from the left by

\gamma^\nu\left(\mathrm{i}\partial_\nu - \frac{e}{c}A_\nu\right) + mc

and, by using the commutation relations for the

\gamma

matrices, bring the expression obtained into quadratic form in terms of the field tensor

\left[\left(\mathrm{i}\hbar\partial - \frac{e}{c}A\right)^2 - \frac{\hbar e}{2c}\sigma^{\mu\nu}F_{\mu\nu} - m^2c^2\right]\psi = 0\,.

The assertion follows by evaluating the expression

\sigma^{\mu\nu}F_{\mu\nu}

using the explicit form of the field tensor as a function of the fields

\boldsymbol{E}

and

\boldsymbol{B}