Advanced Solid State Physics

4 Quantization of the Electromagnetic Field - Quantization Recipe

This chapter serves as a template for the quantization of phonons, magnons, plasmons, electrons,
spinons, holons and other quantum particles that inhabit solids. It is also shown how macroscopic,
easier measurable properties are derived.

The first aim is to get to the equation of motion of electrodynamics - the wave equation. In electro-
dynamics Maxwell’s equations describe the whole system:

∇·E=

ρ

 0

(27)

∇·B= 0 (28)

∇×E=−

∂B

∂t

(29)

∇×B=μ 0 j+μ 0  0

∂E

∂t

(30)

The field can also be described by a scalar potentialV and a vector potentialA:

E=−∇V −

∂A

∂t

B=∇×A

In vacuum,ρ= 0andj= 0, therefore alsoV can be chosen to vanish. Coulomb gauge (∇·A= 0) is
considered to get

∇·

∂A

∂t

= 0

∇·∇×A= 0

∇×

∂A

∂t

=

∂

∂t

∇×A

∇×∇×A=−μ 0  0

∂^2 A

∂t^2

(31)

The first three equations don’t help because they are just a kind of vector identities. But with eqn.
(31) and the vector identity

∇×∇×A=∇(∇·A)−∇^2 A=−∇^2 A

because of Coulomb gauge the wave equation

c^2 ∇^2 A=

∂^2 A

∂t^2

(32)

can be derived. Target reached - now the next step can be done.

The wave equation can be solved by normal mode solutions, which have the form

A(r,t) =Acos(k·r−ωt)