(3 - hard) Electrons have spin 1/2. This means that their wave function comes back to itself after a rotation by 4π. In the cyclotron motion, what happens to the wave function phase after one full orbit around B in physical space: did the phase rotate by 2π or by 4π, or by another value?
Attempt of an answer: Along a path in a magnetic field with vector potential $A$, the phase $\delta$ acquired by the wave function of a charged particle is $(e/\hbar) \int A dx$. 246
After one complete orbit, the phase change is therefore given by $\delta= e \Phi / h = e B \pi r^2 / h$, where $r$ is the radius of the orbit and where $h= 2 \pi \hbar$. Using the cyclotron radius $r=m v / eB$ we appear to get $$\delta = \pi (m v)^2/(e B h)$$
I hope this is correct! This result of the phase $\delta$ acquired by an orbiting electron thus depends on its speed and on the magnetic field. In any case, the relation for the orbital phase change is not a simple, fixed multiple of $\pi$, as we could have wished for.
(2 - simple) Is it correct to say that in spin up state, therefore the spin points always away from the axis above the rotation plane, and for spin down it always points away below the rotation plane? (Assuming "above" is where the magnetic field B points to.)
This question should be answered "yes" by a graph from hyperphysics:
In the spin up state, the spin points always away from the axis and above the rotation plane, and for spin down it always points below the rotation plane and outside.
(1 - simple) (A) Is it correct to say that the electron(s), when looking in the direction of the field, orbit(s) with/along the clock, due to the negative charge? (B) And that the spin precesses in the same direction/sense?
(A) This should be answered "yes" by this graph from hyperphysics on the cyclotron, which is for positively charged particles though:
Indeed, the electron(s), when looking in the direction of the field, orbit(s) clockwise.
(B) The question on precession for negatively charged particles should be answered "yes" by this picture from wikipedia, where the large red arrow is the magnetic field:
The sense of precession is the same as the sense of rotation. 649