A spin-one-half magnetic moment sits in a magnetic field of fixed magnitude
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations
The Structure of Quantum Mechanics, Ch Slow change, sudden change and the Berry phase
Problem
A spin-one-half magnetic moment sits in a magnetic field of fixed magnitude. The field direction n^ makes a constant angle β=30∘ with the z-axis and is swept slowly once around the z-axis, so its azimuthal angle φ runs from 0 to 2π while β stays fixed; the field returns to its starting direction. The sweep is slow enough that the adiabatic theorem applies throughout.
((a)) Find the solid angle Ω subtended at the origin by the circuit traced out by n^ in field space.
((b)) Find the geometric (Berry) phase γ+ acquired by the spin state that starts, and ends, as the instantaneous eigenstate aligned with the field (the "spin up along n^" eigenstate).