Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 35C (Principles of Quantum Mechanics)

The spin components satisfyThe irreducible representations are indexed byand have orthonormal basis $|s,\sigma ⟩$, whereOn this basis,Each Cartesian component is Hermitian,and $S_{+}^{†}=S_{-}$. AlsoRotational equivalence, or the off-diagonal form of $S_x,S_y$ in this basis, gives

Use the ordered basisThe ladder coefficients are $\sqrt{3},2,\sqrt{3}$, soThereforeand

The required state is the maximal-spin coherent state pointing at azimuth $\phi$ in the equatorial plane. In the same ordered $S_z$ basis it isIts norm is $(1+3+3+1)/8=1$. Direct multiplication using the matrices in part (b), or rotation of $|\hat{\mathbf{z}},3/2⟩$, givesIt therefore yields the value $3\hslash /2$ with probability one when spin is measured along $\mathbf{n}$.

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 $-\gamma B$.