 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2016/paper-1/4/b/solution

Work in temporal gauge, with zero scalar potential, and use the charge sign exactly as in the given covariant spatial derivative. A spatially uniform vector potential acts on a Fourier mode as

(\nabla - \frac{i e A ( t )}{ℏ}) {ψ_{k} (t) e^{i k \cdot r}} = i (k - \frac{e A ( t )}{ℏ}) ψ_{k} (t) e^{i k \cdot r} .

A

. Substitution into the field equation and multiplication by

c^{2}

therefore gives

\ddot{ψ}_{k} + ω_{k - e A (t) /ℏ}^{2} ψ_{k} = 0, ω_{q} = c^{2} ∣ q ∣^{2} + \frac{m ^{2} c ^{4}}{ℏ ^{2}} .

The canonical wavevector

k

ℏ k - e A (t)

. The absence of a term involving

\dot{A}

in this mode equation follows from temporal gauge and the spatial minimal-coupling operator. The electric field still affects evolution through the changing oscillator frequency.

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