Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 23

about its center of mass. The quantizedenergy levels of a diatomic in two

dimensions are

with degeneracies for Jnot equal to zero, and whenJ = 0.

As usual, whereI is the moment of inertia.

Hint: For getting out of the wizard’s evil clutches, treat all levels as having

the samedegeneracy andthen.... Oh, no!He’s got me, too!

Assuming derive the partition function for an individ-

ual diatomic molecule in two dimensions.

Determine the thermodynamic energy E and heat capacity in the

limit, where for a set of indistinguishable,independent,

heteronuclear diatomic molecules constrained to rotate in a plane.

Compare these results to those for an ordinary diatomic rotor in three

dimensions. Comment on the differences and discuss briefly in terms

of the number of degrees of freedom required to describe the motion

of a diatomic rotor confined to a plane.

Diatomic Molecules in Three Dimensions (Stony

Brook, Michigan State)

4.44

Consider thefreerotation of adiatomicmolecule consisting of two atoms
of mass and respectively, separated by adistance Assumethat
the molecule is rigid with center of mass fixed.

a) Starting from the kinetic energy where

derive the kinetic energy of this system in spherical coordinates and

show that

whereIis the moment of inertia. ExpressIin terms of and

Derive the canonical conjugate momenta and Express the

Hamiltonian of thissystem interms of and I.

The classical partitionfunction is defined as

b)

c)

a)

b)