A2.14

Quantum Physics
Part II, 2003

(i) A system of NN distinguishable non-interacting particles has energy levels EiE_{i} with degeneracy gi,1i<g_{i}, 1 \leqslant i<\infty, for each particle. Show that in thermal equilibrium the number of particles NiN_{i} with energy EiE_{i} is given by

Ni=gieβ(Eiμ)N_{i}=g_{i} e^{-\beta\left(E_{i}-\mu\right)}

where β\beta and μ\mu are parameters whose physical significance should be briefly explained.

A gas comprises a set of atoms with non-degenerate energy levels Ei,1i<E_{i}, 1 \leqslant i<\infty. Assume that the gas is dilute and the motion of the atoms can be neglected. For such a gas the atoms can be treated as distinguishable. Show that when the system is at temperature TT, the number of atoms NiN_{i} at level ii and the number NjN_{j} at level jj satisfy

NiNj=e(EiEj)/kT\frac{N_{i}}{N_{j}}=e^{-\left(E_{i}-E_{j}\right) / k T}

where kk is Boltzmann's constant.

(ii) A system of bosons has a set of energy levels WaW_{a} with degeneracy fa,1a<f_{a}, 1 \leqslant a<\infty, for each particle. In thermal equilibrium at temperature TT the number nan_{a} of particles in level aa is

na=fae(Waμ)/kT1n_{a}=\frac{f_{a}}{e^{\left(W_{a}-\mu\right) / k T}-1} \text {. }

What is the value of μ\mu when the particles are photons?

Given that the density of states ρ(ω)\rho(\omega) for photons of frequency ω\omega in a cubical box of side LL is

ρ(ω)=L3ω2π2c3\rho(\omega)=L^{3} \frac{\omega^{2}}{\pi^{2} c^{3}}

where cc is the speed of light, show that at temperature TT the density of photons in the frequency range ωω+dω\omega \rightarrow \omega+d \omega is n(ω)dωn(\omega) d \omega where

n(ω)=ω2π2c31eω/kT1n(\omega)=\frac{\omega^{2}}{\pi^{2} c^{3}} \frac{1}{e^{\hbar \omega / k T}-1}

Deduce the energy density, ϵ(ω)\epsilon(\omega), for photons of frequency ω\omega.

The cubical box is occupied by the gas of atoms described in Part (i) in the presence of photons at temperature TT. Consider the two atomic levels ii and jj where Ei>EjE_{i}>E_{j} and EiEj=ωE_{i}-E_{j}=\hbar \omega. The rate of spontaneous photon emission for the transition iji \rightarrow j is AijA_{i j}. The rate of absorption is Bjiϵ(ω)B_{j i} \epsilon(\omega) and the rate of stimulated emission is Bijϵ(ω)B_{i j} \epsilon(\omega). Show that the requirement that these processes maintain the atoms and photons in thermal equilibrium yields the relations

Bij=BjiB_{i j}=B_{j i}

and

Aij=(ω3π2c3)BijA_{i j}=\left(\frac{\hbar \omega^{3}}{\pi^{2} c^{3}}\right) B_{i j}