In the case of a cyclotron, with a constant magnetic field $B$ in the vertical direction, a moving electron circles in a horizontal orbit.
The cyclotron frequency is $\omega = eB/m$.
At the same time, the spin precesses around the magnetic field with the Larmor frequency. (I assume $g=2$ here, so neglect the anomalous magnetic moment.) For $g=2$, the Larmor frequency is the same as the cyclotron frequency, if I am not wrong. Therefore the spin points always away from the axis of rotation (which is the magnetic field direction). I understand that such experiments are regularly made with many electrons circling in ring accelerators.
Now here are my simple and my hard questions:
(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?
(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.)
(3 - hard) Electrons have spin 1/2. This means that their wave function comes back to itself after a rotation by $4 \pi$. In the cyclotron motion, what happens to the wave function phase $\delta$ after one full orbit around $B$ in physical space: did the phase rotate by $2 \pi$ or by $4 \pi$, or by another value?