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Mathematics for Physics

109 Dipole Selection Rules

Quantum-mechanical constraints that are used to decide which single-photon electric-dipole transitions are allowed, where an electric-dipole transition is a jump driven by the operator \(e\mathbf{r}\).

\(\Delta\ell=\pm 1\). The electric-dipole matrix element vanishes unless the orbital quantum number changes by one unit. This principle is used to forbid \(\Delta\ell=0\) lines in the electric-dipole approximation.

The orbital selection rule is

\[ \Delta\ell = \pm 1 \]

where

  • \(\ell\) is the orbital quantum number.

\(\Delta m=0,\pm 1\). The magnetic quantum number can change by \(0\) or \(\pm 1\). This principle is used to assign polarizations to the components of a line.

The magnetic selection rule is

\[ \Delta m = 0,\pm 1 \]

where

  • \(m\) is the magnetic quantum number.

The parity-change rule for electric-dipole radiation. The parity of the wavefunction must change. Parity is the even or odd character of a wavefunction under inversion. This principle is used to forbid transitions between two even states or two odd states.

109.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — dipole selection rules.
  2. Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. — electric-dipole transitions.