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Journal of the Korean Physical Society

Spin behavior in a quarter rotation of the magnetic field

Hitoshi Kuwabara, Yasushi Yamauchi, Andrew Pratt

Published May 1, 2013• 2 citations
Publisher

Abstract

The spin rotation of a particle after a quarter rotation of an external magnetic field is calculated numerically and discussed both in the adiabatic and the nonadiabatic schemes. Adding to the well-known fully adiabatic case, we find that the spin rotates a quarter turn along the magnetic field’s direction when the transient time of the magnetic field matches closely but not exactly the Larmor precession period or its multiples.

DOI
10.3938/jkps.62.1286

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The spin rotation of a particle after a quarter rotation of an external magnetic field is calculated numerically and discussed both in the adiabatic and the nonadiabatic schemes. Adding to the well-known fully adiabatic case, we find that the spin rotates a quarter turn along the magnetic field’s direction when the transient time of the magnetic field matches closely but not exactly the Larmor precession period or its multiples.
We examine the problem of calculating the spin up and spin down amplitudes for a spin 12 particle, initially aligned with respect to a constant magnetic field, and then subjected to a rotation of the field through an arbitrary angle in a finite time. We find that the solutions to the problem are entirely determined by an adiabaticity parameter α which is the ratio of the particle Larmor frequency to the field rotation rate. As expected, a spin flip occurs readily only in the nonadiabatic case, α≪1.
It is known that the action of a magnetic field on the electron leads to the precession of its spin. However, if we describe this effect within the Minkowski space, then it turns out that the result of the action is not the precession of spin of electron, but the rotation of the electron with the angular velocity –(e/mc)H, which corresponds to the experimental data (e and m are the charge and mass of the electron, c is the speed of light, H is a magnetic field strength). Bibliogr. 7 references.
It is well known that any cyclic solution of a spin 1/2 neutral particle moving in an arbitrary magnetic field has a nonadiabatic geometric phase proportional to the solid angle subtended by the trace of the spin. For neutral particles with higher spin, this is true for cyclic solutions with special initial conditions. For more general cyclic solutions, however, this does not hold. As an example, we consider the most general solutions of such particles moving in a rotating magnetic field. If the parameters of the system are appropriately chosen, all solutions are cyclic. The nonadiabatic geometric phase and the solid angle are both calculated explicitly. It turns out that the nonadiabatic geometric phase contains an extra term in addition to the one proportional to the solid angle. The extra term vanishes automatically for spin 1/2. For higher spin, however, it depends on the initial condition. We also consider the valence electron of an alkaline atom. For cyclic solutions with special initial conditions in an arbitrary strong magnetic field, we prove that the nonadiabatic geometric phase is a linear combination of the two solid angles subtended by the traces of the orbit and spin angular momenta. For more general cyclic solutions in a strong rotating magnetic field, the nonadiabatic geometric phase also contains extra terms in addition to the linear combination.
The probability amplitude for a spin-1/2 particle in a precessing magnetic field to be in an energy eigenstate is calculated exactly. The probability for the particle to be in a given energy eigenstate and the corresponding phase of the amplitude are obtained. In the adiabatic limit there are no transitions, and the phase is the sum of the dynamical phase and the Berry phase. When a unitary transformation is made, the Hamiltonian and the energy operator transform differently. The energy eigenvalues, Berry phases, and probability amplitudes calculated in a frame rotating with the magnetic field are the same as in the laboratory frame. If the energy operator is incorrectly taken to be the Hamiltonian in the rotating frame, incorrect energy eigenvalues, Berry phases, and probability amplitudes are obtained. Nevertheless, in the adiabatic limit the correct total phase results.
With the help of the time-dependent gauge transformation technique, we have studied the geometric phase of a spin-half particle in a rotating magnetic field. We have found that the slow but finite frequency of the rotating magnetic field will make the difference between the adiabatic geometric phase and the exact geometric phase. When the frequency is much smaller than the energy space and the adiabatic condition is perfectly guaranteed, the adiabatic approximation geometric phase is exactly consistent with the adiabatic geometric phase. A simple relation for the accuracy of the adiabatic approximation is given in terms of the changing rate of the frequency of the rotating magnetic field and the energy level space.