Most forces in nature depend only on distance: gravity, the Coulomb force, the average force a nucleus exerts. A potential $V(r)$ is unchanged by every rotation about its centre, so it commutes with every component of angular momentum. Then the energy, $\hat{L}^{2}$ and $\hat{L}_{z}$ can all be known together, and every stationary state is a function of $r$ times a spherical harmonic. The angles are done once and for all; only the radial part is left.

A lighthouse lamp spreads the same beam in every direction around it. Describing its light needs only the distance from the lamp; the direction takes care of itself. A central potential leaves the angles to the spherical harmonics in the same way.

Spherical coordinates, with the $z$ axis pointing up: the distance $r$ from the centre, the angle $\theta$ from the $z$ axis, and the angle $\varphi$ round it, not drawn. A central potential depends on $r$ alone, so the dependence on and is always a spherical harmonic.

For $V=V(r)$, the Hamiltonian commutes with $\hat{L}^{2}$ and $\hat{L}_{z}$, and the three share their eigenstates.

• $[\hat{H},\hat{L}_{i}]=0$ for every component, and $$: angular momentum is conserved.

$\hat{L}_{z}$ turns momenta and positions round the $z$ axis, which leaves $\hat{p}^{2}$ and the distance unchanged. So it commutes with both parts of the Hamiltonian; the other components follow by symmetry.

The same pattern as $[\hat{L}_{z},\hat{x}]=i\hbar\hat{y}$ and .

The two terms cancel; $\hat{p}_{z}$ commutes with $\hat{L}_{z}$ outright.

$\hat{L}_{z}=-i\hbar\,\partial_{\varphi}$, and does not depend on .

Both parts of the Hamiltonian commute with $\hat{L}_{z}$.

No axis is special for $V(r)$, so the same holds for every component and for $\hat{L}^{2}$.

In a central potential the Hamiltonian, $\hat{L}^{2}$ and $\hat{L}_{z}$ commute, so each stationary state can be labelled by its energy, its and its .

The energy of a central potential cannot depend on $m$: the lowering operator commutes with $\hat{H}$ and moves between the $2\ell+1$ values of $m$ without changing the energy.

A central potential conserves angular momentum but not momentum: pushing the particle sideways changes its distance from the centre.

Every stationary state of a central potential can be written as a radial function times a spherical harmonic.

Great work! You now know how to:

• prove that a central potential conserves angular momentum
• label its stationary states by $E$, $\ell$ and $m$
• explain why energies do not depend on $m$