Exact phases and probabilities for a spin-1/2 particle in a precessing magnetic field

Abstract

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. © 1990 The American Physical Society.