Past exam of the quantum theory course of the University of Oxford / 2023 / 3 / b

Choose the ordered angular momentum eigenstates $(|1,1⟩,|1,0⟩,|1,-1⟩)$, so $J_3=\hslash \mathrm{diag}(1,0,-1)$. Since $[J_3,J_{\pm }]=\pm \hslash J_{\pm }$, a nonzero matrix element of $J_{+}$ must increase the magnetic quantum number by one, and one of $J_{-}$ must decrease it by one. Thus only entries $(1,2),(2,3)$ of $J_{+}$ and $(2,1),(3,2)$ of $J_{-}$ can be nonzero. The adjoint operator identity $J_{-}=J_{+}^{†}$ givesFor the $j=1$ irreducible spin representation, both adjacent transitions are present; the following calculation determines their sizes.

The phases of the three basis vectors can be chosen so that $A$ and $B$ are nonnegative real numbers. Matrix multiplication then givesThe angular momentum commutation relations require this to equal $2\hslash J_3=2{\hslash }^2\mathrm{diag}(1,0,-1)$. Equating diagonal entries yieldsHenceThese are also the $j=1$ special case of the normalized spin ladder operator coefficients $\hslash \sqrt{j(j+1)-\mu (\mu \pm 1)}$.

Recover the angular momentum operators from $J_1=(J_{+}+J_{-})/2$ and $J_2=(J_{+}-J_{-})/(2i)$. The spin-one angular momentum matrices in the chosen descending-$J_3$ orthonormal basis areThey are self-adjoint operators, satisfy $[J_i,J_j]=i\hslash {\epsilon }_{ijk}{J}_k$, and give $J^2=2{\hslash }^{2}I$, as required for $j=1$.