Consider the open rotation of a diatomic molecule consisting of two atoms of mass and respectively divided by a distance presume that the molecule is rigid with center of mass fixed.

a) Starting from the kinetic energy ε_k, where

Obtain the kinetic energy of this system in spherical coordinates and show that

Where I is the moment of inertia. Express I in terms of m1, m2 and a.

b) Derive the canonical conjugate momenta P θ and P? Express the Hamiltonian of this system in terms of P θ, P?, ?, ?, and I.

Calculate Z_cl. Calculate the heat capacity for a system of N molecules.

d) Presume now that the rotational motion of the molecule is described by quantum mechanics. Mark the partition function in this case taking into relation of the degeneracy of each state. Compute the heat capacity of a system of N molecules in the limit of low and high temperatures and compare them to the classical result.