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Question: a twodimensional quantum rotor has quantum states m with energy...

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A two-dimensional quantum rotor has quantum states m with energy Em 2I where I is a constant moment of inertia and m is any integer. The angular momentum of state m is l hm. Consider a system of N distinguisable rotors. The rotors are able to exhange energy and angular momentum, allowing those quantities to come to equilibrium. Suppose that you know the total energy of the system is U and the total angular momentum in the system is L. Analyze the system using a rotating canonical ensemble Follow the derivation of the canonical ensemble, but introduce an additional Lagrange multiplier to enforce L Xm mlm, where n is the number of particles in state m. Determine the most probable value for n in terms of Em, lm, and the various Lagrange multipliers. Define the fugacity z in the usual way (a) Use the resulting distribution function to express N, U, and Las functions of 2, B and Assume the relevant sums can be approximated as integrals. (b) Express U as a function of N, T, and l L/N. Interpret your result (c) Determine the physical significance of the parameter. Hint: what, classically, is l/I equal to? This illustrates how the canonical ensemble method can be extended to incorporate other con- served quantities.

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