[Physics] When an electron orbits in a magnetic field, how exactly does its spin precess

Question

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?

Answer 1

(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 phase after one full orbit around B in physical space: did the phase rotate by 2π or by 4π?

The rotation of the synchrotron is around a macroscopic axis, the wavefunction's 2 or 4π rotation and consequent phase is around an axis through the center of mass of the electron. The macroscopic rotation is not relevant.

If one wants to go into details there will be phases that could be modeled quantum mechanically, but no longer simple one electron states. Have a look to get at the complexity here.

Edit after comment by OP:

But in the cyclotron, the rotation of an electron around its own axis and the motion around the orbit are coupled. So the integral over the vector potential A can be taken. What is the result? That is what I want to know.

The electron is a quantum entity, and the phase is a phase within the Ψ wavefunction of the electron, which is a function of (x,y,z,t) and the energy momentum of the fourvector of the electron. In the probability distribution which is the complex conjugate squared of the wavefunction there are no phases. Phases have a measurable effect only in interference terms, otherwise they are irrelevant specialized experiments have to be designed to "measure" the phase.