Specific Heat of Diatomic Gas Rotations.

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Find the specific heat capacity, for the following:
(a) An ideal diatomic gas undergoing rotation with no vibrations

(b) An ideal diatomic gas undergoing rotation with radial vibrations.

(a) We choose the z axis as the axis of rotation and as the angular coordinate about z.
Using Lagrangian mechanics and as our general coordinate. The kinetic energy T is:

So our Hamiltonian is:

We now look for eigenstates of the angular momentum squared quantum ...