Show that the radiation field is transverse

[It's me]

Homework Statement

Show that the radiation field is transverse, $\vec{\nabla}\cdot\vec{A}=0$ and obeys the wave equation $\nabla^2\vec{A}-\frac{1}{c^2}\partial_t^2\vec{A}=0$. You should start from the expansion of the quantum Electromagnetic field.

Homework Equations

$H=\frac{1}{2}\int d^3x(E^2+B^2)$

The Attempt at a Solution

I know that the transverse vector potential gives rise to the EM radiation from moving charges. In this case the Coulomb gauge can be used and both mathematical conditions are met. However, I don't understand what the expansion of the quantum Electromagnetic field or how I could get the answer from it.

 

[Cryo]

Strange exercise. Your initial words are about the radiation field (IMHO electric/magnetic fields), but your equations are about the vector potential. Also, you don't need to go to quantum optics to prove the transversality of EM-fields in the abscence of sources.

If we focus at the first part, expanding field here probably means that you have to define a cavity of some sort and expand your fields in terms of the modes of that cavity. See Loudon's The Quantum Theory of Light, for example. When you do that, you will end up with

$H=\sum modes\, of\, the\, cavity$

i.e. your integral over fields, will become a sum over modes.

 

[Cryo]

At that point, you can quantize the electromagnetic field