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Mathematics for Physics

147 Spontaneous Emission

A radiative transition in which an excited atom drops to a lower energy state by emitting a photon that is used to empty an upper level without an applied field.

The spontaneous frequency condition. An excited atom can emit a photon of energy \(hf=E_{i}-E_{f}\) with no external stimulus. This principle is used to explain ordinary fluorescence and the natural lifetime of an excited state.

The spontaneous-emission condition is

\[ hf = E_{i}-E_{f} \]

where

  • \(h\) is Planck’s constant.
  • \(f\) is the frequency of the emitted photon.
  • \(E_{i}\) and \(E_{f}\) are the upper and lower energies.

The Einstein \(A\) decay. The spontaneous rate is the Einstein \(A\) coefficient. This principle is used to write an exponential decay of the upper-level population.

The spontaneous decay law is

\[ \dfrac{dN_{i}}{dt} = -A_{if}N_{i} \]

where

  • \(N_{i}\) is the number of atoms in the upper level.
  • \(A_{if}\) is the Einstein \(A\) coefficient.
  • \(t\) is time.

The \(A\)–\(B\) relation. The \(A\) coefficient is related to the \(B\) coefficient by the Planck spectrum of thermal radiation. This principle is used to compute lifetimes from absorption strengths.

147.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — spontaneous emission.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — emission of a photon.