Turning a state about the $z$ axis is another continuous family, and its generator is found the same way: turn by a tiny angle, see what the wavefunction does, and read off the operator. It is $\hat{L}_{z}=\hat{x}\hat{p}_{y}-\hat{y}\hat{p}_{x}$, the $z$ component of angular momentum. Rotations about different axes do not commute — a book turned about two axes in two orders ends up differently — and that failure is written into the commutators of the three components, which the next chapter builds everything on.

A record on a turntable is rotated a little each moment; knowing how each point moves in one small turn — at right angles to its radius, in proportion to its distance — fixes the whole rotation. Angular momentum is that small turn, written as an operator.

A turn by a small angle $\varepsilon$ about the $z$ axis moves the point $\mathbf{r}=(x,y)$ along its circle, at right angles to $\mathbf{r}$: $\delta \mathbf{r}=$, drawn here with . On wavefunctions that displacement is generated by .

Rotating a state by an angle $\varphi$ about the $z$ axis gives the wavefunction $\psi(R^{-1}\mathbf{r})$. The rotations about one axis form a family whose generator is the component of angular momentum along it.

• About any axis $\mathbf{n}$: $\hat{R}_{\mathbf{n}}(\varphi)=e^{-i\varphi\,\mathbf{n}\cdot\hat{\mathbf{L}}/\hbar}$ with .

A rotated state takes at $\mathbf{r}$ the value the old one had at the point rotated back. For a small angle that point is $(x+\varepsilon y,\,y-\varepsilon x)$; the first-order Taylor expansion produces $x\partial_{y}-y\partial_{x}$, which is divided by .

Rotating back by a small angle $\varepsilon$.

Taylor’s theorem to first order in $\varepsilon$.

With $\hat{p}_{x}=-i\hbar\,\partial_{x}$ and .

Since $x\partial_{y}-y\partial_{x}=\frac{i}{\hbar}\hat{L}_{z}$.

A finite rotation is many small ones.

Rotating a state by a small angle $\varepsilon$ about the $z$ axis subtracts $(i\varepsilon/\hbar)\hat{L}_{z}$ applied to it, where .

In the angle $\varphi$ around the $z$ axis, $\hat{L}_{z}$ is $-i\hbar$ times the derivative with respect to , so is an eigenfunction with eigenvalue .

A book turned by a right angle about $x$ and then about $y$ ends up differently from one turned in the other order. The generators of rotations about different axes do not commute either.

An eigenstate of $\hat{L}_{z}$ with eigenvalue $m\hbar$ is only multiplied by $e^{-im\varphi}$ when rotated by about the axis.

Great work! You now know how to:

• derive $\hat{L}_{z}$ as the generator of rotations about the $z$ axis
• use $\hat{L}_{z}=-i\hbar\,\partial/\partial\varphi$ and its eigenfunctions