To summarize the result of the calculations of the last section we have the **Hamiltonian for the radiation field**.

By now we know that if the and do not commute, neither do the and so we should continue to avoid commuting them.

Since we are dealing with harmonic oscillators, we want to find the analog of the **raising and lowering operators**.
We developed the raising and lowering operators by trying to write the Hamiltonian as
.
Following the same idea, we get

So these are definitely the raising and lowering operators. Of course the commutator would be zero if the operators were not for the same oscillator.

(Note that all of our commutators are assumed to be taken at equal time.) The Hamiltonian is written in terms and in the same way as for the 1D harmonic oscillator. Therefore, everything we know about the raising and lowering operators applies here, including the commutator with the Hamiltonian, the raising and lowering of energy eigenstates, and even the constants.

The can only take on

As with the 1D harmonic oscillator, we also can define the **number operator**.

The last step is to **compute the raising and lowering operators in terms of the original coefficients**.

Jim Branson 2013-04-22