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 volume
and apply periodic boundary conditions on our electromagnetic waves.
We also assume for now that there are no sources inside the region so that we can make a gauge transformation
We decompose the field into its Fourier components at
We know the time dependence of the waves from Maxwell's equation,
Note again that we have made this a transverse field by construction. The unit vectors are transverse to the direction of propagation. Also note that we are working in a gauge with , so this can also represent the 4-vector form of the potential. The Fourier decomposition of the radiation field can be written very simply.
Let's verify that this decomposition of the radiation field satisfies the Maxwell equation, just for some practice. Its most convenient to use the covariant form of the equation and field.
Let's also verify that .
Jim Branson 2013-04-22