In 1868 astronomers pointed their spectroscopes at the Sun during an eclipse and found a bright yellow line that matched no element known on Earth. They named the mystery substance after the Greek word for the Sun — helios — and it was another twenty-seven years before helium was finally isolated in a laboratory. We had read a brand-new element off the face of a star we can never touch, using nothing but the colours of its light. That is the astonishing power of atomic spectroscopy: every atom carries a barcode of sharp, discrete lines, unique to that element, printed in the light it emits and absorbs.

You have seen these barcodes yourself. The flat orange glow of a sodium street lamp is essentially one pair of yellow lines. A neon sign is a fistful of red-orange lines. Fireworks are chemistry set to music — strontium red, barium green, copper blue — each colour a signature set of atomic transitions. This page answers two linked questions: why are the lines discrete (rather than a smooth rainbow), and why do only certain lines appear while others, seemingly available, are missing? The second question is the story of selection rules, and it reaches all the way from the sodium doublet to the 21-centimetre radio glow that maps our galaxy.

Discrete levels make discrete lines

A bound electron cannot have just any energy. Solving the Schrödinger equation for the atom leaves only a discrete ladder of allowed energies — the stationary states. When an electron drops from an upper state of energy E_\text{up} to a lower state E_\text{low}, the surplus energy leaves as a single photon whose frequency f is fixed by the gap:

Because the gaps are sharp, the photon energies — and hence the wavelengths — are sharp too. A continuous rainbow would mean a continuum of energies; the fact that we see thin bright lines is direct evidence that the atom's energies are quantised. Run the logic backwards and you get spectroscopy's superpower: measure the wavelengths, and you have measured the energy-level structure of the atom. Each element's ladder is different, so each element's set of lines is a fingerprint.

Hydrogen — one proton, one electron — is the case we can solve exactly, and its ladder is famously simple. The energy of level n is

The minus sign means the electron is bound: it would take 13.6\ \text{eV} (the ionisation energy) to tear the ground-state electron free to E_\infty = 0. The levels crowd together as n grows, piling up towards the ionisation limit.

A picture of the ladder

The diagram below draws hydrogen's levels to scale (energy running up the page) and then builds up two of its series as downward arrows — one photon emitted per arrow. Notice how tightly the upper levels bunch near the top while the ground state sits far below: that huge n=2 \to n=1 drop is why the Lyman lines are energetic ultraviolet, while the smaller Balmer drops land in the visible. Reveal the series one at a time.

Worked example — a Balmer line

The red hydrogen line. Find the wavelength of the n = 3 \to n = 2 transition. Using the energy ladder,

Convert to a wavelength with the handy shortcut \lambda(\text{nm}) = 1239.8 / \Delta E(\text{eV}):

That is the deep red \text{H}\alpha line — the glow of hydrogen nebulae and the classic first line of the Balmer series. The same arithmetic with n_2 = 4, 5, 6 gives the cyan, violet and near-ultraviolet Balmer lines, all marching towards the series limit at 364.6\ \text{nm}.

Labelling states: the quantum numbers

To say which transitions are allowed we first need to label the states properly. The energy E_n alone is not the whole story — each level carries a set of quantum numbers describing the electron's angular momentum:

• Principal n = 1, 2, 3, \dots — sets the energy (in hydrogen) and the overall size of the orbital.
• Orbital angular momentum \ell = 0, 1, \dots, n-1 — the shape; spectroscopists write these as letters s, p, d, f, \dots
• Magnetic m_\ell = -\ell, \dots, +\ell — the orientation, giving 2\ell + 1 sub-states.
• Spin s = \tfrac12 with m_s = \pm\tfrac12 — the electron's intrinsic angular momentum.

Orbital and spin angular momentum add (vectorially) into a total angular momentum \mathbf{J} = \mathbf{L} + \mathbf{S}, whose quantum number J runs in integer steps from |L - S| to L + S. Physicists package all of this into a term symbol {}^{2S+1}L_J: the left superscript is the spin multiplicity 2S+1, the letter is the total orbital L (S, P, D, F…), and the subscript is J. For example a single electron in a p state (L = 1, S = \tfrac12) has J = \tfrac12 or \tfrac32, giving the two terms {}^{2}P_{1/2} and {}^{2}P_{3/2} — remember that pair, it is about to explain the sodium doublet.

Selection rules: why only some lines appear

Here is the puzzle. Hydrogen has levels at n = 1, 2, 3, \dots, so you might expect a photon for every pair. But look closely at real spectra and many "obvious" transitions are missing or absurdly faint. The reason is that emitting a photon is not free — the transition rate is governed by the electric-dipole matrix element \langle f\,|\,\mathbf{r}\,|\,i\rangle, the overlap of the initial and final states weighted by the position operator \mathbf{r}. When this integral is zero, the fastest (electric-dipole, "E1") channel is switched off and the line is called forbidden.

The integral vanishes unless the two states are related in specific ways, and those conditions are the selection rules. Physically they are angular-momentum bookkeeping: the photon carries away spin-1 (one unit of angular momentum) and has odd parity, so the atom's quantum numbers must change by exactly the right amount to balance the books.

The star of the show is \Delta\ell = \pm 1. Notice what it does not say: it places no restriction on \Delta n. An electron can drop from n = 7 to n = 1 in one leap, as long as the orbital quantum number changes by one. What is restricted is the shape, not the shell.

Worked example — allowed or forbidden?

Apply \Delta\ell = \pm 1 to a few hydrogen transitions:

• 2p \to 1s: \ell goes 1 \to 0, so \Delta\ell = -1. Allowed — this is the bright Lyman-\alpha line.
• 2s \to 1s: \ell goes 0 \to 0, so \Delta\ell = 0. Forbidden. The 2s state has no E1 route down and becomes metastable, living about a tenth of a second — an eternity in atomic terms.
• 3d \to 2p: \ell goes 2 \to 1, \Delta\ell = -1. Allowed.
• 3d \to 1s: \ell goes 2 \to 0, \Delta\ell = -2. Forbidden as a single E1 photon — the electron must instead cascade 3d \to 2p \to 1s.

And a term-symbol example: for a {}^{2}P_{3/2} \to {}^{2}S_{1/2} transition, \Delta L = 1 (P→S), \Delta S = 0, and \Delta J = -1 — all satisfied, so it is allowed. This is exactly the sodium D-line, which we meet next.

Fine structure: the sodium doublet

Look at the famous yellow of a sodium lamp through a good spectroscope and the single line splits into two, at 589.0\ \text{nm} and 589.6\ \text{nm} — the sodium D-lines. This tiny splitting is fine structure, and it comes from spin–orbit coupling.

In the electron's own frame, the positively charged nucleus orbits it, making a current loop and hence a magnetic field. The electron's spin magnetic moment sits in that field, and the interaction energy depends on the relative orientation of spin and orbit:

Because \mathbf{L}\cdot\mathbf{S} depends on how L and S combine into J, states with different J split apart in energy. Sodium's outer electron in a 3p state (L = 1, S = \tfrac12) has two possible totals, J = \tfrac12 and J = \tfrac32 — the terms {}^{2}P_{1/2} and {}^{2}P_{3/2}. The decay to the 3s\ ({}^{2}S_{1/2}) ground state therefore has two slightly different energies: the doublet.

The scale of fine structure is set by the fine-structure constant \alpha \approx 1/137: the splittings go as \alpha^2 \approx 5\times 10^{-5} times the gross energy, which is why the two D-lines differ by only 0.6\ \text{nm} out of 589. Two further layers sit beneath: hyperfine structure, an even smaller splitting from the coupling to the nuclear spin (the origin of the 21 cm line), and the Zeeman effect, in which an external magnetic field splits the m_\ell (and m_s) sub-levels — turning one line into several and letting us measure the magnetic fields of sunspots and distant stars.