Rotate a state of definite $j$ and it stays within the same $j$: rotations commute with $\hat{J}^{2}$, so they only reshuffle the $2j+1$ states of one ladder among themselves. The reshuffling is a $(2j+1)\times(2j+1)$ unitary matrix, the rotation matrix. About the $z$ axis it is diagonal — each state just acquires a phase $e^{-im\varphi}$. About another axis it mixes the $m$ values, with probabilities that are fixed functions of the angle. And for half-integer $j$ a whole turn gives $-1$, a sign no ordinary object shows.

Turning a box of coloured beads changes which colour faces up, but never the number or kind of beads in the box. A rotation changes the mixture of $m$ values but never $j$.

A state $\ket{1,0}$ rotated by an angle $\beta$ about the $y$ axis, from 0 to $\pi$: the probability of still finding $m=0$ is , and of finding or is each. The three always add to one, and at the state is back to .

A rotation $\hat{R}=e^{-i\beta\,\mathbf{n}\cdot\hat{\mathbf{J}}/\hbar}$ maps into combinations of the states of the same . The coefficients form a unitary matrix .

• About the $z$ axis the matrix is diagonal: $D^{(j)}_{m'm}=e^{-im\varphi}\delta_{m'm}$.

$\hat{J}^{2}$ commutes with every component, so with every function of them, including any rotation. A rotated eigenstate of $\hat{J}^{2}$ is therefore an eigenstate with the same eigenvalue, which is a combination of the states of that .

From the lesson on commutators.

A power series in the components.

Every term commutes with $\hat{J}^{2}$.

The rotated state has the same eigenvalue of $\hat{J}^{2}$.

Expand it in the states of that $j$; the coefficients are the rotation matrix.

A rotation commutes with $\hat{J}^{2}$, so it maps the states of one $j$ among themselves. Only the mixture of $m$ values changes.

Rotating about the $z$ axis only multiplies each $\ket{j,m}$ by the phase $e^{-im\varphi}$.

For half-integer $j$, a rotation by a full turn multiplies every state by $-1$. Two full turns are needed to come back exactly.

Rotating $\ket{1,0}$ by $\beta$ about the $y$ axis leaves $m=0$ with probability $$ and gives with probability each.

Great work! You now know how to:

• explain why rotations keep $j$ and mix $m$
• use rotation matrices about the $z$ and $y$ axes
• show that half-integer states change sign under a full turn