A Spin Flip Problem

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.