Statistical Mechanics · Problem 2.2

The Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · phase-space-volume-element

Problem

(a) Verify explicitly the invariance of the volume element

d\omega

of the phase space of a single particle under transformation from the Cartesian coordinates

(x, y, z, p_x, p_y, p_z)

to the spherical polar coordinates

(r, \theta, \phi, p_r, p_\theta, p_\phi)

. (b) The foregoing result seems to contradict the intuitive notion of ``equal weights for equal solid angles'', because the factor

\sin\theta

is invisible in the expression for

d\omega

. Show that if we average out any physical quantity, whose dependence on

p_\theta

and

p_\phi

comes only through the kinetic energy of the particle, then as a result of integration over these variables we do indeed recover the factor

\sin\theta

with the sub-element

(d\theta\, d\phi)