selection rules
An atom has a ladder of energy levels, but not every pair of rungs is connected by a bright spectral line. Most conceivable transitions simply do not happen — their radiation rate is zero by symmetry. Selection rules are the bookkeeping that tells you in advance which transitions are allowed and which are forbidden, so you know which lines a spectrum can show.
They come from the transition matrix element that controls the rate. For the dominant electric-dipole (E1) process the rate is proportional to |<f| e r |i>|^2, and this integral vanishes unless the initial and final states differ in a specific way. Evaluating the angular part (via the Wigner-Eckart theorem, essentially integrals of three spherical harmonics) gives, for a one-electron jump: Delta l = ±1, Delta m_l = 0, ±1, and the parity of the state must change. Including spin and total angular momentum: Delta j = 0, ±1 (but j = 0 to j = 0 is forbidden), Delta m_j = 0, ±1, and in LS coupling Delta S = 0 with Delta L = 0, ±1. Each rule is the statement that a particular symmetry (rotational invariance, parity) makes the dipole integral zero unless it is respected.
Selection rules are the grammar of spectroscopy — they explain which lasers can lase on which transitions, why some excited states are metastable, and how to read a spectrum. The essential caveat: 'forbidden' means forbidden at the electric-dipole level, not physically impossible. Transitions that break the E1 rules can still proceed through weaker higher-order channels — magnetic-dipole (M1) or electric-quadrupole (E2) radiation — typically slower by factors of (alpha)^2 or more. The hydrogen 21-cm line and many astrophysical 'forbidden lines' are exactly such M1/E2 transitions, faint per atom but visible over cosmic quantities of gas.
Hydrogen's 2s to 1s transition is E1-forbidden: both states have l = 0 (so Delta l = 0, not ±1) and the same even parity. The 2s state therefore cannot decay by a single dipole photon and is metastable, living about 0.1 s and decaying mainly by rare two-photon emission — an eternity compared with the ~1.6 ns lifetime of the E1-allowed 2p state.
A forbidden transition is not impossible — it is just slow, which makes metastable states.
Selection rules are stated per multipole and per coupling scheme; a transition 'forbidden' for E1 may be 'allowed' for M1 or E2, so always say which multipole and whether you are in LS or jj coupling before quoting a rule.