Selection Rules for Transitions

Hydrogen has infinitely many energy levels, so naively any pair could exchange a photon. In practice only certain transitions occur. Selection rules tell us which jumps are allowed and which are forbidden, and they explain the observed pattern of spectral lines.

Transitions come from a matrix element

When light interacts with an atom, the leading coupling is the electric dipole interaction, proportional to the position operator $\mathbf{r}$. The rate of a transition between an initial state $|n\ell m\rangle$ and a final state $|n'\ell'm'\rangle$ is governed by the dipole matrix element

If this integral vanishes, the transition is electric-dipole forbidden: it does not occur at the leading order. The selection rules are simply the conditions under which $\mathbf{d}_{fi} \ne 0$.

The angular selection rules

The position operator $\mathbf{r}$ has the angular character of an $\ell = 1$ object (its components are proportional to $Y_1^m$). Combining it with the spherical harmonics of the states and using their orthogonality gives the electric-dipole selection rules:

The $\Delta\ell = \pm 1$ rule reflects conservation of angular momentum: the absorbed or emitted photon carries one unit of angular momentum, so $\ell$ must change by exactly one. Notice that $\Delta\ell = 0$ is forbidden — an $s \to s$ transition cannot happen by a single dipole photon. The $\Delta m$ rule corresponds to the three polarizations of the photon ($m$ unchanged for light polarized along $z$, $\Delta m = \pm 1$ for circular polarizations in the plane).

What about $n$?

There is no restriction on $\Delta n$. The principal quantum number can change by any amount; the radial integral $\int R_{n'\ell'} r R_{n\ell}\,r^2 dr$ is generically nonzero. So $3p \to 1s$, $2p \to 1s$, and $4p \to 2s$ are all allowed, while $2s \to 1s$ (which would need $\Delta\ell = 0$) is dipole-forbidden.

Why the rules matter

Selection rules explain why only certain spectral lines appear and why some excited states are metastable. The hydrogen $2s$ state cannot decay to $1s$ by a single dipole photon (it would need $\Delta\ell = 0$), so it lives far longer than the $2p$ state, decaying instead by a slow two-photon process. The same logic governs lasers, fluorescence, and the forbidden lines seen in astrophysical nebulae, where low densities let metastable states finally radiate.

The takeaway

Electric-dipole transitions in hydrogen require $\Delta\ell = \pm 1$ and $\Delta m = 0, \pm 1$, with no constraint on $\Delta n$. These rules follow from the $\ell = 1$ character of the position operator and conservation of angular momentum, and they determine which spectral lines are observed.

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