Classically, angular momentum is $\mathbf{r}\times\mathbf{p}$: a vector at right angles to the plane of the motion, as long as the distance times the momentum across it. In quantum mechanics its three components become operators, each the generator of rotations about its axis. They are observables, but not compatible ones: no state has a definite value of all three unless all three are zero. What can be known together is the total, $\hat{L}^{2}$, and one component, by convention $\hat{L}_{z}$.

A spinning top has an axis you can point to. A quantum top has a definite length of angular momentum and a definite projection on one axis, but the rest of its direction is as uncertain as a position and a momentum are.

A particle going round a circle. Its angular momentum $\mathbf{L}=\mathbf{r}\times\mathbf{p}$ points out of the page and has size $rp$. In quantum mechanics its three components become operators, and they cannot all be known at once.

Replace $\mathbf{r}$ and $\mathbf{p}$ in $\mathbf{L}=\mathbf{r}\times\mathbf{p}$ by operators. Each component is Hermitian, and the square of the whole vector is a fourth observable.

• Each factor in a product such as $\hat{y}\hat{p}_{z}$ belongs to a different direction, so the factors commute and the order does not matter.
• ${\stackrel{}{L}}_{}$, and generate rotations about their axes, as the symmetry chapter found for .

The adjoint of a product reverses its order. In $\hat{L}_{x}=\hat{y}\hat{p}_{z}-\hat{z}\hat{p}_{y}$ each product joins a coordinate to a momentum in another direction, which commute, so reversing the order changes nothing.

The $x$ component of $\hat{\mathbf{r}}\times\hat{\mathbf{p}}$.

The adjoint reverses products, and positions and momenta are Hermitian.

A coordinate commutes with the momentum along a different axis.

Swap each pair back; the same holds for the other two components.

The components of angular momentum do not commute with one another, but each commutes with $\hat{L}^{2}$. So the length of the angular momentum and one component can be known together; two components cannot.

In $\hat{L}_{x}=\hat{y}\hat{p}_{z}-\hat{z}\hat{p}_{y}$ each coordinate is multiplied by a momentum along another axis. Those commute, so the products need no ordering rule, and each component is Hermitian.

For a particle going round a circle of radius $r$ with momentum $p$ along the circle, the angular momentum points along the axis and has size $rp$.

The convention is to choose $\hat{L}^{2}$ and $\hat{L}_{z}$ as the observables with definite values. Nothing is special about : any direction could be chosen instead.

Great work! You now know how to:

• write the components of angular momentum as operators
• show that they are Hermitian
• say which angular momentum observables can be known together