Thermodynamics, Statistical Physics, and Quantum Mechanics

97

4.13 Maxwell–Boltzmann Averages (MIT)

a) We may write the unnormalizedMaxwell–Boltzmanndistributionimme-

diately as

We would like towrite(S.4.13.1) as so we must integrateover all

velocities inorder to find the proper normalization:

Rewriting (S.4.13.2) in spherical coordinates we have

A variety of problems contain thedefiniteintegral(S.4.13.3) and its varia-

tions. A particularly easy way to derive it is to start by writing the integral

as

Now multiplyI by itself, replacing by yielding

Rewriting(S.4.13.5) inpolar coordinatesgives

where wehavesubstituted in(S.4.13.6). So wehave
Integratinginsteadfrom 0 to thengives

THERMODYNAMICS AND STATISTICAL PHYSICS