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Mathematics for Physics

92 Absorption

A quantum transition in which a system takes energy from an electromagnetic field that is used to raise the system from a lower level to a higher level, where a photon is a discrete packet of light that disappears in the process.

The absorption frequency condition. Absorption occurs when a photon of energy \(hf\) matches the gap \(E_{f}-E_{i}\) with \(E_{f}>E_{i}\). This principle is used to compute the frequency of an absorption line.

The absorption condition is

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

where

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

The Einstein \(B\) rate. The absorption rate is proportional to the energy density of the radiation and to the Einstein \(B\) coefficient. This principle is used to write the number of absorptions per unit time.

The absorption rate is

\[ W_{\mathrm{abs}} = B_{if}\rho(f) \]

where

  • \(W_{\mathrm{abs}}\) is the absorption rate.
  • \(B_{if}\) is the Einstein \(B\) coefficient.
  • \(\rho(f)\) is the radiation energy density at frequency \(f\).

The link to stimulated emission. The same matrix element that governs absorption governs stimulated emission. This principle is used to relate \(B_{if}\) and \(B_{fi}\).

92.1 References

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