We now have the **Hamiltonian for the radiation field**.

We can then

This now

We wish to write the Hamiltonian in terms of a coordinate for each oscillator and the conjugate momenta.
The coordinate should be real so it can be represented by a Hermitian operator and have a physical meaning.
The simplest choice for a real coordinates is
.
With a little effort we can **identify the coordinate**

and its

The

This verifies that this choice gives the right Hamiltonian. We should also check that this choice of

This one checks out OK.

The **other equation of Hamilton** is

This also checks out, so

We have a collection of uncoupled oscillators with identified canonical coordinate and momentum. The next step is to quantize the oscillators.

Jim Branson 2013-04-22