130_notes.dvi

We will now study the radiation field in a region with no sources so that∇·~ A~= 0. We will use the
equations

B~ = ∇×~ A~

E~ = −^1

c

∂A~

∂t

∇^2 A~−

1

c^2

∂A~

∂t^2

= 0

33.2 Fourier Decomposition of Radiation Oscillators

Our goal is to write the Hamiltonian for the radiation field in terms of a sum of harmonic oscillator
Hamiltonians. The first step is to write the radiation field in as simple a way as possible, as a sum
of harmonic components. We will work in a cubic volumeV=L^3 and applyperiodic boundary
conditionson our electromagnetic waves. We also assume for now that there areno sources inside
the regionso that we can make a gauge transformation to makeA 0 = 0 and hence∇·~ A~= 0. We
decompose the field into its Fourier componentsatt= 0

A~(~x,t= 0) =√^1

V

∑

k

∑^2

α=1

ˆǫ(α)

(

ck,α(t= 0)ei

~k·~x

+c∗k,α(t= 0)e−i

~k·~x)

where ˆǫ(α) are real unit vectors, andck,αis the coefficient of the wave with wave vector~kand
polarization vector ˆǫ(α). Once the wave vector is chosen, the two polarization vectors must be
picked so that ˆǫ(1), ˆǫ(2), and~kform a right handed orthogonal system. The components of
the wave vector must satisfy

ki=

2 πni

L

due to the periodic boundary conditions. The factor out front is set to normalize the states nicely
since
1
V

∫

d^3 xei

~k·~x

e−i

~k′·~x

=δ~k~k′

and
ˆǫ(α)·ˆǫ(α

′)

=δαα′.

We know thetime dependence of the wavesfrom Maxwell’s equation,

ck,α(t) =ck,α(0)e−iωt

whereω=kc. We can now write thevector potential as a function of position and time.

A~(~x,t) =√^1

V

∑

k

∑^2

α=1

ǫˆ(α)

(

ck,α(t)ei

~k·~x

+c∗k,α(t)e−i

~k·~x)

We may write this solution in several different ways, and use the bestone for the calculation being
performed. One nice way to write this is in terms 4-vectorkμ, the wave number,

kμ=

pμ

̄h

= (kx,ky,kz,ik) = (kx,ky,kz,i

ω

c

)