Light drives an atom from one level to another through the electric dipole operator $e\hat{\mathbf{r}}$: a jump from $\ket{i}$ to $\ket{f}$ goes at a rate proportional to $|\bra{f}\hat{\mathbf{r}}\ket{i}|^{2}$. Most of these matrix elements vanish for reasons of symmetry alone. Parity: $\hat{\mathbf{r}}$ is odd, so the two levels must have opposite parity. Angular momentum: $\hat{\mathbf{r}}$ is a vector, which carries one unit, so $\ell$ can change by at most one — and with parity, by exactly one — while $m$ changes by 0 or $\pm1$. These are the dipole selection rules. In hydrogen $2p\to1s$ is allowed and $2s\to1s$ is forbidden, and 2s lives about a hundred million times longer as a result.

A key opens only locks cut to its shape. The dipole operator carries exactly one unit of angular momentum and odd parity, and it can join only levels whose angular momenta and parities differ in just that way.

The lowest levels of hydrogen, set in columns by $\ell$ with the energies not to scale, and the dipole jumps between them: every arrow changes $\ell$ by one. There is no arrow from 2s, which can reach 1s only by far slower processes.

Between states $\ket{n\ell m}$ of a central potential, the dipole matrix element vanishes unless

• The rate of a jump driven by light is proportional to $|\bra{f}\vec{\epsilon}\cdot\hat{\mathbf{r}}\ket{i}|^{2}$, with the polarisation; this lesson asks only when that element can be nonzero.

Use commutators with $\hat{L}_{z}$ to find which changes of $m$ each component allows, parity to rule out an even change of $\ell$, and the addition rule, applied to the angular parts of the functions, to rule out a change of $$ by more than one.

Sandwich the commutator between the two states.

The same, for the components that raise and lower $m$.

$\hat{\mathbf{r}}$ is odd and the states have parities $(-1)^{\ell}$ and $(-1)^{\ell'}$.

The angles of $\hat{\mathbf{r}}$ are a $j=1$ set; by the addition rule, applied to angular functions as to states, $1$ and $\ell$ make only these.

Parity removes $\ell'=\ell$, leaving a change of exactly one.

A dipole jump changes $\ell$ by exactly one and $m$ by at most one.

The dipole operator is odd under inversion. Its matrix element between two states of the same parity is the integral of an odd function, which is zero.

The component $x+iy$ raises $m$ by one, $x-iy$ lowers it by one, and $z$ leaves it alone.

Light polarised along $z$ couples through the $z$ component of $\mathbf{r}$, which commutes with $\hat{L}_{z}$.

Great work! You now know how to:

• derive the dipole selection rules from parity and angular momentum
• tell allowed from forbidden jumps in hydrogen
• say why forbidden levels live long