A2.14
(i) A system of distinguishable non-interacting particles has energy levels with degeneracy , for each particle. Show that in thermal equilibrium the number of particles with energy is given by
where and are parameters whose physical significance should be briefly explained.
A gas comprises a set of atoms with non-degenerate energy levels . 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 , the number of atoms at level and the number at level satisfy
where is Boltzmann's constant.
(ii) A system of bosons has a set of energy levels with degeneracy , for each particle. In thermal equilibrium at temperature the number of particles in level is
What is the value of when the particles are photons?
Given that the density of states for photons of frequency in a cubical box of side is
where is the speed of light, show that at temperature the density of photons in the frequency range is where
Deduce the energy density, , for photons of frequency .
The cubical box is occupied by the gas of atoms described in Part (i) in the presence of photons at temperature . Consider the two atomic levels and where and . The rate of spontaneous photon emission for the transition is . The rate of absorption is and the rate of stimulated emission is . Show that the requirement that these processes maintain the atoms and photons in thermal equilibrium yields the relations
and