144 Selection Rules

Quantum-mechanical constraints that are used to decide which transitions between energy states are allowed, where an allowed transition is a transition whose matrix element is not zero.

The nonzero-matrix-element criterion. A transition is allowed only if the matrix element of the perturbation between the two states is nonzero. This principle is used to discard jumps that violate a symmetry.

The transition matrix element is

\[ M_{fi} = \langle f|\hat{H}'|i\rangle \]

where

\(M_{fi}\) is the matrix element.
\(\hat{H}'\) is the perturbation.
\(|i\rangle\) and \(|f\rangle\) are the initial and final states.

The electric-dipole selection rules. Electric dipole radiation changes the orbital quantum number by one unit and changes \(m\) by \(0\) or \(\pm 1\). This principle is used to write the dipole selection rules.

The electric-dipole selection rules are

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

where

\(\ell\) is the orbital quantum number.
\(m\) is the magnetic quantum number.

Forbidden transitions from conserved angular momentum and parity. Transitions that would violate conservation of angular momentum or parity are forbidden. Parity is the even or odd character of a wavefunction under inversion. This principle is used to explain missing lines in a spectrum.

144.1 References

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

143 Schrodinger Equations

145 Spin