The adiabatic approximation says that if H′ changes slowly, the system remains in the n-th instantaneous eigenstate. But suppose the Hamiltonian changes slowly but by a large amount. For instance, suppose a spin-1/2 particle starts out in the spin-up state (m=+1/2), in a strong field pointing in the z-direction; then the field slowly rotates, and finally points in the −z-direction.
(a) If ∣B∣ is constant, and the direction changes from z^ to −z^ slowly and uniformly (in latitude, not in azimuthal angle), what is the probability of a spin flip at the end?
(b) If the field starts at B0z^ (at t=0) and decreases to zero (at t=T) slowly, what is the probability that the particle is in the spin-down state at time T?