A proton's spin (a spin one-half system) precesses at Larmor frequency in a static field

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Spin one-half, SU(2) and two-level dynamics

Problem

A proton's spin (a spin one-half system) precesses at Larmor frequency

f_0

in a static field. Starting in the state

|{\uparrow}\rangle

(spin up along the static field), it is driven by a radiofrequency field of fixed amplitude, corresponding to a Rabi frequency

\omega_1/2\pi = 50\ \mathrm{kHz}

, in two successive stages. ((a)) In the first stage, lasting

t_a=2.0\ \mu\mathrm{s}

, the drive frequency is offset from

f_0

\Delta f_a=5.0\ \mathrm{MHz}

. Find an upper bound on the probability of finding the spin in

|{\downarrow}\rangle

at the end of this stage, and show it is negligible. ((b)) In the second stage, lasting

t_b=5.0\ \mu\mathrm{s}

, the drive frequency is switched to exact resonance with

f_0

. Taking the spin to still be in

|{\uparrow}\rangle

at the start of this stage (justified by part (a)), find the probability of measuring it in

|{\downarrow}\rangle

at the end of the second stage.