Spin-one maximum-projection eigenvector ($v_{+}=({\cos }^2(\theta /2),\sin \theta /\sqrt{2},{\sin }^2(\theta /2){)}^{\mathsf{T}}$)

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.