Fermions in a two-level system (with degeneracy)
Consider a system of N independent fermions. Assume that the single-particle Hamiltonians have only two energy levels, with energies 0 and ?. However, the two levels have degeneracies n0 and n1, which are, of course, both integers.
1. First take the simple case of n0 = n1 = 1, with N = 1. Find the chemical potential, μ, as a function of temperature. What is the Fermi energy, ?F = μ(T = 0)?
2. Now make it more interesting by taking arbitrary values of n0 and n1, but specifying that N = n0. Again find the chemical potential, μ, as a function of temperature for low temperatures. That is, assume that β? » 1. What is the Fermi energy?
3. Keep arbitrary values of n0 and n1, but consider the case of N? » 1. What is the Fermi energy?
4. Keep arbitrary values of n0 and n1, but consider the case of N>n0. Again find the chemical potential, μ, as a function of temperature for low temperatures. That is, assume that β? » 1. What is the Fermi energy?