Statistical Physics, Second Revised and Enlarged Edition

Summary 49

i.e.

ε=(h^2 / 8 Ma^2 )(n^21 +n^22 +n^23 ) (4.8)

The energy(h^2 / 8 Ma^2 )=εεj,say, gives a scale for the energy level spacing for a
helium atominthegas. It works out at 8× 10 −^40 J, a verysmallenergycompared
withkkkBTwhich equals 4 × 10 −^21 Jat 300 K. ClearlykkkBTεεj,and from what we
know already about energy scales this implies that a very large number of states will
be energeticallyavailablefor occupationinthethermalequilibrium state. Hence the
whole approach of this chapter.
Finally we derive the density of states. Helium is a monatomic gas, so (4. 5 ) gives
thedensityofstatesink.In thisinstance, since^4 Hehas zero spin, thefactorG=1.
Henceg(k)is known. The appropriate dispersion relation (from the Schrödinger
equation)is (4.7), so thedensity ofstatesinεcanbedetermined, treatingkand
thereforeεas continuous variablesinview ofthelarge numbers ofstatesinvolved.
The calculation goes as follows. Starting from (4.7),ε=^2 k^2 / 2 M,differentiation
gives

δε =^2 kδk/M

and inversiongives

k=( 2 Mε/)^1 /^2

Equation (4. 5 ) withG=1is

g(k)δk=V/( 2 π)^3 · 4 πk^2 δk

Substitutingforkandforkδkwe obtain

g(ε)δε=V/( 2 π)^3 · 4 π( 2 Mε/^2 )^1 /^2 (Mδε/^2 )

=V 2 π( 2 M/h^2 )^3 /^2 ε^1 /^2 δε (4.9)

Thisderivation nicely demonstrates the transformationfromktoεandthefinalresult
(4.9) will be a useful one in our later discussions ofgases.

4 .4Summary

Thischapterlaysthegroundworkforlaterdiscussion ofgases,by discussingthe
one-particle states.

1. 
   

   Since a gas particle is free to roam over a whole macroscopic box, its possible
   states are very closely spaced.

   
   


2. 
   

   Thegeometricalideas of‘fittingwavesintoboxes’ applyto anygaseous system.