Quantum statistical mechanics: A many-spin system (This is a more difficult version of the previous problem)
Consider a macroscopic crystal with a spin-one quantum mechanical magnetic moment located on each of N atoms. Assume that we can represent the energy eigenvalues of the system with a Hamiltonian of the form
where each σn takes on the values -1, 0, or +1, and B and D are constants representing an external magnetic field and a "crystal field," respectively. The entire system is in contact with a thermal reservoir at temperature T.
1. Calculate the partition function for this system.
2. Calculate the free energy of this system.
3. Calculate the magnetization per spin
4. Calculate the entropy per spin of this system.
5. Determine whether this system satisfies the Nernst Postulate for all values of the parameters.