Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 35C / d / Solution

Write $\omega =\gamma B$. Sincethe time-evolution operator is

For the initial state $|\hat{\mathbf{z}},3/2⟩=|3/2⟩$,Only an overall phase changes, so every spin measurement has time-independent probabilities.

For the initial state along $\hat{\mathbf{x}}$, part (c) with $\phi =0$ givesUp to the overall phase $e^{3i\omega t/2}$, this is the coherent state of part (c) with $\phi =-\omega t$. Thus the spin direction undergoes Larmor precession about the $z$-axis with angular velocity $-\gamma B$.

Solved by gpt-5.6-sol high.