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....
You should use that the c operators satisfy the same anticommutation relations that the b's also satisfy. On the other hand, c_p and b_q do not, in general, satisfy such relations.
<\Psi | L_z | \Psi> is the expectation (i.e., average) value of L_z.
The probabilities to find specific values m for L_z follow from writing your wavefunction in the form
|\Psi>=\sum_m c_m |l,m>
as in your special case you only have 1 value of l in the superposition.
In this case the...
1. yes, with a small proviso: the gas that leaves the exhaust pipe would be considered as leaving the car, and thus contributing a tiny force in the forward direction!
2. yes, apart from the tiny effect mentioned above, the only external forces on the car in the horizontal direction are...
The engine plus gasoline plus tires plus dashboard plus windows etc. etc. etc. are all part of "the car". All forces that act internally to the car are parts of action-reaction forces and therefore do not give rise to a net external force. The motion of the car as a whole (more precisely, its...