Spin coherent state

A spin coherent state is obtained by rotating the maximal-weight state $|s,s⟩$ so that it is an eigenstate of spin along a chosen direction with eigenvalue $s\hslash$.

Prepare the $J_1=+\hslash$ spin coherent state and evolve with Hamiltonian operator $H=\omega J_3$ for time $t$. In the descending-$J_3$ basis of the spin-one angular momentum matrices, the initial vector is $(1,\sqrt{2},1{)}^{\mathsf{T}}/2$ and the evolved vector is $(e^{-i\theta },\sqrt{2},e^{i\theta }{)}^{\mathsf{T}}/2$, where $\theta =\omega t$. The $J_1$ orthonormal basis has vectors $(1,\sqrt{2},1{)}^{\mathsf{T}}/2$, $(1,0,-1{)}^{\mathsf{T}}/\sqrt{2}$ and $(1,-\sqrt{2},1{)}^{\mathsf{T}}/2$. Taking their inner products with the evolved vector gives probability amplitudes $(1+\cos \theta )/2$, $-i\sin \theta /\sqrt{2}$ and $(\cos \theta -1)/2$. Their squared moduli give the Born rule probabilities in the title. At integer half-turns $\theta =\pi k$, the outcome is deterministically $(-1{)}^k\hslash$.

In the ordered z-projection basis $|1,1⟩,|1,0⟩,|1,-1⟩$, the normalized eigenvector of $J_x\sin \theta +J_z\cos \theta$ with eigenvalue $+\hslash$ is the displayed vector. It follows by a y-axis rotation of the maximum-weight state, or direct solution of the tridiagonal eigenvalue equation. Squaring its entries gives the three z-projection measurement probabilities.

A spin coherent-state path integral inserts group-averaged resolutions of the identity between short time steps. Its kinetic term is the Spin coherent-state Berry phase, equal to $S$ times the oriented solid angle swept on the sphere.

At azimuth $\phi$, the normalized spin-$3/2$ coherent-state components are proportional to $1,\sqrt{3}{e}^{i\phi },\sqrt{3}{e}^{2i\phi },e^{3i\phi }$.