For j=1/2 there are two states: m=+/- 1/2, and for j=3/2 there are 4 different m-states.
The "weak" Zeeman effect just refers to a situation where the energy shift due to the magnetic field is small and can be treated with perturbation theory: the unperturbed Hamiltonian has split the l=1 level...
It's because the interaction that splits the energies of the state is the spin-orbit coupling, proportional to \vec{L}\cdot\vec{S}, which can be rewritten as being proportional to the difference \vec{J}^2-\vec{L}^2-\vec{S}^2, which is dependent only on the quantum numbers j and l (s=1/2 in...
You clearly did not state the full problem so I have to keep guessing: were you supposed to diagonalize the Hamiltonian and find U such that [tex] H=\sum_k E(k) b^+_kb_k [/itex]?
If the b's are fermionic annihilation operators, then that *means* they satisfy the anticommutation relations that, as you figured out, are equivalent to U being unitary. Done.
You need more information to prove any of those relations. You must have been given some info about what the b's are supposed to be, for instance. I assumed that you had been told that the b's are fermionic annihilation operators.
There are (at least) two sorts of average you can take:
If you want to calculate a force averaged over *distance*, you can use the change in kinetic energy, divided by distance: Work =\delta E=F_average * d
If you want to calculate an average over *time*, then that would be given by the change...
No, you have to prove U is unitary.
Edit: you already seem to know that U being unitary is equivalent to the b's satisfying the same anticommutation relations as the c's. But that's all there is to it....