Selection rules and spectra

Spectral lines

Atoms emit and absorb light at discrete wavelengths because their energies are quantized. When an atom transitions from a higher energy state to a lower energy state, it emits a photon with energy

$hf=E_i-E_f$

For absorption, the atom takes in a photon and moves to a higher state if the photon energy matches the energy difference.

Hydrogen's spectral lines were known before quantum mechanics and helped motivate atomic theory.

Hydrogen transition energies

Using

E_n=-rac{13.6 eV}{n^2}

transition photon energy is

ight)$$ for emission from $n_i$ to $n_f$ with $n_i>n_f$. The wavelength satisfies $$rac{hc}{lambda}=E_i-E_f$$ Hydrogen spectral series include Lyman transitions ending at $n=1$, Balmer transitions ending at $n=2$, and Paschen transitions ending at $n=3$. ## Rydberg formula The wavelengths of hydrogen spectral lines follow $$rac{1}{lambda}=R_Hleft(rac{1}{n_f^2}-rac{1}{n_i^2} ight)$$ where $R_H$ is the Rydberg constant for hydrogen. This empirical formula was explained by quantized energy levels. ## Why selection rules exist Not every mathematically possible energy difference produces a strong transition. Light interacts with atoms primarily through the electric dipole interaction. Transition probability depends on a matrix element such as $$langle f|hat{ec{r}}|i angle$$ If this matrix element is zero because of symmetry, the electric dipole transition is forbidden or strongly suppressed. ## Electric dipole selection rules For hydrogen-like atoms, common electric dipole selection rules are $$Delta l=pm1$$ and $$Delta m=0,pm1$$ These rules come from angular momentum conservation and parity properties of the dipole operator. The photon carries angular momentum, so atomic angular momentum quantum numbers must change consistently. ## Allowed and forbidden transitions Allowed transitions are relatively likely and produce strong spectral lines. Forbidden transitions are not absolutely impossible in all circumstances; they may occur through weaker magnetic dipole, electric quadrupole, or multi-photon processes. They are just forbidden in the leading electric dipole approximation. Forbidden lines can be important in low-density astrophysical gases, where excited states survive long enough to decay through weak channels. ## Spectroscopy as a probe Spectral lines reveal atomic composition, temperature, density, motion, magnetic fields, and electric fields. Doppler shifts reveal motion. Zeeman splitting reveals magnetic fields. Stark shifts reveal electric fields. Quantum mechanics turns spectra into a diagnostic tool for laboratories, stars, galaxies, and plasmas. ## The big idea Atomic spectra arise from transitions between quantized states. Photon energies match level differences through $hf=E_i-E_f$. Selection rules determine which transitions are strong, with electric dipole rules such as $Delta l=pm1$ and $Delta m=0,pm1$. Spectroscopy is one of quantum mechanics' great experimental successes.