We **decompose the radiation field into its Fourier components**

where are real unit vectors, and is the coefficient of the wave with wave vector and polarization vector . Once the wave vector is chosen, the two polarization vectors must be picked so that , , and

Plugging the Fourier decomposition into the formula for the Hamiltonian density and using the transverse nature of the radiation field, we can compute the Hamiltonian (density integrated over volume).

This Hamiltonian will be used to quantize the EM field. In calculating the Hamiltonian, care has been taken not to commute the Fourier coefficients and their conjugates.

The canonical coordinate and momenta may be found

for the harmonic oscillator at each frequency. We assume that a coordinate and its conjugate momentum have the same commutator as in wave mechanics and that coordinates from different oscillators commute.

As was done for the 1D harmonic oscillator, we write the Hamiltonian in terms of raising and lowering operators that have the same commutation relations as in the 1D harmonic oscillator.

This means everything we know about the raising and lowering operators applies here. Energies are in steps of and there must be a ground state. The states can be labeled by a quantum number .

The Fourier coefficients can now be written in terms of the raising and lowering operators for the field.

States of the field are given by the occupation number of each possible photon state.

Any state can be constructed by operating with creation operators on the vacuum state. Any state with multiple photons will automatically be symmetric under the interchange of pairs of photons because the operators commute.

This is essentially the same result as our earlier guess to put an in the emission operator.

We can now write the quantized radiation field in terms of the operators at .

Beyond the Electric Dipole approximation, the next term in the expansion of
is
.
This term gets split according to its rotation and Lorentz transformation properties into the Electric Quadrupole term
and the Magnetic Dipole term.
The interaction of the **electron spin** with the magnetic field is of the same order and should be included
together with the E2 and M1 terms.

The Electric Quadrupole (E2) term **does not change parity** and gives us the selection rule.

The Magnetic Dipole term (M1) does not change parity but may change the spin. Since it is an (axial) vector operator, it changes angular momentum by 0, +1, or -1 unit.

The quantized field is very helpful in the derivation of Plank's black body radiation formula that started the quantum revolution. By balancing the reaction rates proportional to and for absorption and emission in equilibrium the energy density in the radiation field inside a cavity is easily derived.

Jim Branson 2013-04-22