# Overlap between eigenstates of angular momentum operators

Consider the states $$\left|j,m_x\right>_x$$ and $$\left|j,m_z\right>_z$$ with total angular momentum $$j$$ and the angular momentum operators $$\hat{S}_x$$ and $$\hat{S}_y$$. In particular, assume that the states satisfy $$\hat{S}_x\left|j,m_x\right>_x=m_x\left|j,m_x\right>_x$$ and $$\hat{S}_z\left|j,m_z\right>_z=m_z\left|j,m_z\right>_z$$. I am wondering about the closed form of the overlap between the eigenstates shown, that is, what is the analytical expression for $$_x\left_z$$.

• $S$, $L$, $J$ are usually spin, orbit, total angular momentum, with quantum numbers $|s, s_z\rangle$, $|l, m\rangle$, $|j, j_z\rangle$ or $3$ in place of $z$.
– JEB
Commented Sep 2, 2023 at 0:48

The transformation that takes you from $$S_z$$ to $$S_x$$ is a rotation about $$\hat y$$ by $$\pi/2$$ so all eigenstates of $$S_x$$ are of the form $$\vert jm\rangle_x=R^{-1}_y(\pi/2)\vert jm\rangle_z$$ so that $$S_xR^{-1}_y(\pi/2)\vert jm\rangle_z= [R^{-1}_y(\pi/2) S_z R_y(\pi/2)]R^{-1}_y(\pi/2)\vert jm\rangle_z=m R^{-1}_y(\pi/2)\vert jm\rangle_z$$ Thus, the closed form is $$_x\langle jm’\vert jm\rangle_z =_z\langle jm’\vert R_y(\pi/2)\vert jm\rangle_z=d^{j}_{m’m}(\pi/2)$$ where $$d$$ is a Wigner little-d function.
(I hope I didn’t mess up the sign between $$\pi/2$$ and $$-\pi/2$$ but aside from this the answer is correct.)