The vector potential can be in terms of a superposition of cavity modes

Quantization of the radiation field

The particle view of radiation is illustrated in this section. We show that photons can be treated mathematically in the same way we treat phonons (or vibrations) in the harmonic oscillator problem. There is a certain logic to this the classical view of electromagnetic radiation and vibrations both represent oscillatory motion. Although the term radiation field sounds abstract, we are being asked to picture the radiation as a number of incident photons impinging upon an area. This is analogous to the concept of flux. However, instead of energy per unit area, now we can count the number of photons per unit area.

The vector potential can be in terms of a superposition of cavity modes. The wavevector k = w_k/c and the frequencies w_k = 2pn_k. The polarization is e_k.

The classical radiation hamiltonian is

In the wave picture

The superposition of cavity modes leads to

Inserting these expressions into the radiation hamiltonian gives

Upon squaring the polarization will drop out. Recalling the k = w/c we now have

The time dependent terms do not depend on r, and the spatial integral is

so the volume term will cancel leaving

This hamiltonian has the same form as the harmonic oscillator. Recall that the force constant k (not to be confused with the wavevector k we have been using thus far) is given by

Thus, a harmonic oscillator hamiltonian can be written

The radiation field hamiltonian is the same except for the absence of the mass term.

In analogy with the harmonic oscillator problem we can define raising and lowering operators

Just as was done for the harmonic oscillator, we can derive commutation relations for the raising and lowering operators of the radiation field. We begin with the position and momentum commutator

Consider a state that has n_k photons. The effect of the raising operator is to create a photon.

Therefore, b_k⁺ is called a creation operator. The effect of the lowering operator is destroy a photon.

Therefore the operator b_k is also called the annihilation operator.

Again the formal analogy between the harmonic oscillator problem and the radiation hamiltonian leads to the following form.

Note that b_k⁺b_kis the number operator. It can also be called n_kand it gives the number of photons. The energy levels are quantized exactly like those of the harmonic oscillator. The zero-point energy is included to allow for spontaneous emission.