Principles of Quantum Mechanics · Problem 18.14

Time-Dependent Perturbation Theory — R. Shankar

Physics — Quantum Mechanics · adiabatic-approximation

Problem

In the adiabatic approximation, the Berry phase is

\gamma_n(t) = i \int_0^t \langle n(t') | \dot{n}(t') \rangle \, dt'.

Note that this is a real number (Problem 18.16). For the case of a spin-1/2 particle in a rotating magnetic field,

\mathbf{B} = B_0(\sin\alpha\cos\omega t, \sin\alpha \sin\omega t, \cos\alpha)

, compute the Berry phase accumulated in one complete cycle (

t: 0 \to 2\pi/\omega

). (a) For the spin-up state. (b) For the spin-down state.