Spin 1/2 rotation with a magnetic field: the phase change from the potential

In the Feynman lectures, an electron beam is turned through an angle. The usual way to do that probably is by applying a magnetic field.

                              ○        ○        ○

                              ○        ○        ○

     e- ●  --->

                              ○        ○        ○

                            magnetic field B

Let us use a magnetic field B whose lines of force are to the direction of the z axis. The z axis points up from the screen. We prepare a coherent electron beam whose spin-z points up. We want to turn the electron beam through 360 degrees using the magnetic field.

The magnetic field introduces a potential to the electrons, and the potential causes a phase shift to the electron wave function.

Let the electrons be non-relativistic, and their velocity v. Let the magnetic field strength be B.

The force is

       F = e v B.

We set F equal to the centrifugal "force":

       m v² / r = e v B

  <=>

       r = m v / (e B).

The length of the loop is 2 π r.

The magnetic moment of the electron to the direction of the z axis is approximately one Bohr magneton:

       μ = e ħ / (2 m).

The potential energy due to the magnetic moment is

       V = μ • B.

Let the kinetic energy of the electrons be much larger than V:

       E = p² / (2 m).

Because of the potential, the momentum p of the electrons in the magnetic field changes by a factor

       1 + 1/2 V / E

       = 1 + 1/2 e ħ B / p².

The de Broglie wavelength of the electron is

       λ = h / p.

The loop 2 π r contains

       n = 2 π p / (e B) * p / h

          = p² / (e ħ B)

wavelengths if we ignore the potential V.

Introducing the potential V causes a change of

       n * 1/2 V / E

       = p² / (e ħ B) * 1/2 e ħ B / p²

       = 1/2

wavelengths. That is, the wave function of the electron changes the sign.

We could explain the sign change of the electron wave function simply by the potential which the magnetic field imposes on the magnetic moment of the electron. There is no need to assume that the spinor is changed in the loop in any way.

"Rotation" around the y axis

Let us assume that we have prepared an electron to the spin-z = +1/2 state. The measuring apparatus points to the direction of the positive z axis.

We then rotate the measuring apparatus through an angle α around the y axis. We do another measurement of the spin.

We do not think it makes sense to speak of this experiment as a "rotation" of the spinor. We did not touch the electron at all and did not rotate its spinor.

Is it a rotation of coordinates? There is no need to rotate the coordinates if we turn the measuring apparatus.

It is a rotation of the measuring apparatus - not of the spinor or coordinates. We prepared the electron in the spin-z +1/2 state, and then measure an observable which does not commute with the prepared state. In particular, if α = 90 degrees, then we measure the spin-x, and have a 50% probability to get +1/2 and 50% for -1/2.

Conclusions

We need to find out if the spin 1/2 spinor rotation algebra in literature is a result of a confusion. People have not taken into account the magnetic potential which can explain a change of the phase in an experiment where a magnetic field is used to "rotate" a particle.

If the "rotation" would be made with an electric field, then the potential can be very small, and we might be able to change the path of an electron with a negligible change of the phase.

This reminds us of our analysis of the Aharonov-Bohm effect on December 21, 2021. Literature forgets that the potential from a magnetic field inside a solenoid does change the phase of an electron when the electron flies past the solenoid.

Classically, a spinning object tries to keep its axis, or precesses if there is a torque. What would the precession mean in quantum mechanics?