A two-dimensional quantum rotor has quantum states m with energy E_m = h^2/2I m^

Question

A two-dimensional quantum rotor has quantum states m with energy E_m = h^2/2I m^2 where I is a constant moment of inertia and m is any integer. The angular momentum of state m is l_m = hm. Consider a system of N distinguishable rotors. The rotors are able to exchange 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 gamma to enforce L = sigma_m n_m l_m, where n_m is the number of particles in state m. Determine the most probable value for n_m in terms of E_m, l_m, 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, Beta and gamma .(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. This illustrates how the canonical ensemble method can be extended to incorporate other conserved quantities.

Explanation / Answer

Solution :-

Given

N = Distinguishable motors

U = Total Energy of the system

L = Total Angular momentum in the system

Quantum rotor has quantum states m with energy

Em = h^2/2I * m^2

Here I = moment of inertia

m is any integer

a) We know that l = L/N

Hence N= L/l

And We know that

l = Lz+ L+ L

Hence N = L/Lz+ L+ L

b) U = Quantum rotor energy + Nl = h^2/2I * m^2 + N*Lz+ L+ L

c) The physical significance of is basically related to l/I = L/NI = Sigma (nm lm)/ NI which symbolize the number of particles in state m and also signifies the most probabale vallue of n(m).

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