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Mathematics for Physics

119 Fermi’s Golden Rule

A quantum-mechanical formula that is used to compute the transition rate per unit time from an initial energy eigenstate into a continuum of final states under a weak constant perturbation.

Fermi’s golden rule. A weak time-independent perturbation \(H'\) induces transitions at a rate proportional to the squared matrix element and to the density of final states. This principle is used to compute decay rates and scattering rates.

Fermi’s golden rule is

\[ \Gamma_{i\to f} = \dfrac{2\pi}{\hbar}\lvert\langle f|H'|i\rangle\rvert^{2}\rho(E_{f}) \]

where

  • \(\Gamma_{i\to f}\) is the transition rate.
  • \(H'\) is the perturbation.
  • \(|i\rangle\) and \(|f\rangle\) are the initial and final states.
  • \(\rho(E_{f})\) is the density of final states at energy \(E_{f}\).
  • \(\hbar\) is the reduced Planck constant.

Energy conservation in the continuum. Energy is conserved within the width set by the time of observation. This principle is used to restrict the final states to those with \(E_{f}\approx E_{i}\).

The harmonic-perturbation form. For a harmonic perturbation of frequency \(\omega\), the same rule applies with \(H'\) the coupling amplitude and with \(E_{f}=E_{i}\pm\hbar\omega\). This principle is used to compute absorption and stimulated emission.

119.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — Fermi’s golden rule.
  2. Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. — time-dependent perturbation theory.