By making use of the adiabatic approximate method, we derive the spin rotation formula of a relativistic neutral particle propagating in a slowly changing magnetic field, which involves the effects of the Berry's phases. In the non–relativistic limit, the Berry's phases result in an additional rotation in OVV spin rotation.
I derive the effective phase of the spin precession for a neutral particle with spin 1∕2 moving in a superposition of constant and radio frequency fields. The fields are perpendicular to each other at all times, and the radio frequency field is slowly rotating with angular speed ω. The derivation is accomplished with the help of the exact solution of the Schrödinger equation. I show that the geometrical (or Berry) phase is the linear term in the power series expansion of the effective phase in terms of ω. I also discuss the role of the Berry phase in the experimental search for an electric dipole moment of the neutron with ultracold neutrons.
By invoking the Rabi rotation technique, the Schrödinger wavefunction of a neutral particle with a magnetic moment and a high-spin is exactly solved in a harmonically-changing magnetic field. This result is then applied to illustrate what the accuracy of the quantum adiabatic approximation is exactly and how the non-adiabatic transition vanishes in accompaniment with the appearence of Berry's geometrical phase. The discussion of this paper verifies C. N. Yang's point of view about the generalization of the quantum adiabatic approximation in connection with Berry's phase.
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.
In this paper we define a non-dynamical phase for a spin-1/2 particle in a rotating magnetic field in the non-adiabatic non-cyclic case, and this phase can be considered as a generalized Berry phase. We show that this phase reduces to the geometric Berry phase, in the adiabatic limit, up to a factor independent of the parameters of the system. We could add an arbitrary phase to the eigenstates of the Hamiltonian due to the gauge freedom. Then, we fix this arbitrary phase by comparing our Berry phase in the adiabatic limit with the Berry's result for the same system. Also, in the extreme non-adiabatic limit our Berry phase vanishes, modulo 2π, as expected. Although, our Berry phase is in general complex, it becomes real in the expected cases: the adiabatic limit, the extreme non-adiabatic limit, and the points at which the state of the system returns to its initial form, up to a phase factor. Therefore, this phase can be considered as a generalization of the Berry phase. Moreover, we investigate the relation between the value of the generalized Berry phase, the period of the states and the period of the Hamiltonian.