ABSTRACT
As noted in Chapter 4, in the classical limit, a large number of eigenstates are populated, albeit in a sparse way, with the mean number of particles in a given state 〈nr 〉 ≪ 1. It is instructive to make use of the familiar classical limit inequality VQ ≪ VA, where VQ = h3/(3mkBT)3/2 is the quantum volume and VA = (V/N) is the volume per particle. e inequality may be rewritten in the following form
N V
h m
k T⎛ ⎝
⎞
⎠
⎛
⎝
⎜
⎞
⎠
⎟
, B
(16.1)
which is convenient for the comparison that is made below. From Equation 4.5, the spacing between adjacent energy levels with quantum numbers (nx, ny, ne) and (nx + 1, ny, nz) is to a good approximation
Δer
xn h mV
≈ ⎛
⎝
⎞
⎠
⎛
⎝
⎜
⎞
⎠
⎟4
(16.2)
Comparison of Equations 16.1 and 16.2 suggests that kBT ≫ Δεr , so that the thermal energy per degree of freedom is very much larger than the spacing of the energy levels, as depicted in Figure 16.1.
