States of the field are given by the occupation number of each possible photon state.
|nk 1 ,α 1 ,nk 2 ,α 2 ,...,nki,αi,...〉=∏
i(a†ki,αi)nki,αi
√
nki,αi!| 0 〉
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.
a†k,αa†k′,α′| 0 〉=a†k′,α′a†k,α| 0 〉This is essentially the same result as our earlier guess to put ann+ 1 in the emission operator (See
Section 29.1).
We can now write the quantized radiation field in terms of the operators att= 0.
Aμ =1
√
V
∑
kα√
̄hc^2
2 ωǫ(μα)(
ak,α(0)eikρxρ+a†k,α(0)e−ikρxρ)
Beyond the Electric Dipole approximation, the next term in the expansion ofei~k·~xisi~k·~x. 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 theelectron spinwith the
magnetic field is of the same order and should be included together with the E2 and M1 terms.
e ̄h
2 mc(~k׈ǫ(λ))·~σThe Electric Quadrupole (E2) termdoes not change parityand gives us the selection rule.
|ℓn−ℓi|≤ 2 ≤ℓn+ℓiThe 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 toNandN+ 1 for
absorption and emission in equilibrium the energy density in the radiation field inside a cavity is
easily derived.
U(ν) =U(ω)dω
dν=
8 π
c^3hν^3
e ̄hω/kT− 11.40 Scattering of Photons
(See section 34) The quantized photon field can be used to computethe cross section for photon
scattering. The electric dipole approximation is used to simplify the atomic matrix element at low
energy where the wavelength is long compared to atomic size.