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# Heat capacity of diatomic molecule in an electric field

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In this problem we'll assume that we are always in the classical regime. This means that the low temperature behavior we want to compute actually means k T much smaller than mu e (e = dipole moment), but k T must not be taken much smaller than hbar^2/(2I), otherwise we need to full quantum mechanical treatment for the angular motion.

We can evaluate the partition function as follows. We know that the partition function for one molecule, Z1, will factorize as follows:

Z1 = Z_{trans} Z_{rot} Z_{ext} (1)

where

Z_{trans} is the contribution from the two dimensional translational motion: energy = 1/(2m) (p_x^2 + p_y^2)

Z_{rot} is the contribution from the rotational motion : energy = 1/(2I) P_theta^2

Z_{ext} is the contribution from the electric field: energy = -mu e cos(theta)

We can compute Z_{trans} as follows:

The number of momentum quantum states in an area A and a range d^2p in momentum space is A d^2p/h^2. Therefore:

Z_{trans} = A/h^2 Integral of d^2p exp[-beta p^2/(2m)] =

A/h^2 Integral of d|p| 2 pi |p| exp[-beta p^2/(2m)]

where the integral over |p| is from zero to infinity. If we evaluate the integral we find the result:

Z_{trans} = A (2 pi m k T)/h^2 (2)

To compute the rotational contribution to Z1, we can use the fact that the rule number of quantum states equals volume in configuration space times volume in momentum space divided by h^(dimension of space) also holds for arbitrary variables if you take th momentum to be the conjugate momentum. In this case if theta is taken as a position variable, then the angular momentum P_{theta} is the conjugate momentum. This means that the number of quantum states between theta and theta + d theta and in a range d P_{theta} in "angular momentum space" is equal to d theta d P_{theta}/h. So, while the integration over theta must be performed when we consider Z_{ext}, de integration measure
d theta actually appears when we compute Z_{rot}.

We ...

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