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Mathematics for Physics

140 Rydberg Formula

A formula that is used to compute the wavelengths of light emitted or absorbed when an electron jumps between shells in hydrogen-like atoms, where a shell is a set of states sharing the same principal quantum number.

The Rydberg formula. The inverse wavelength of a hydrogen line is the Rydberg constant times the difference of inverse squares of two integers. This principle is used to predict the Balmer, Lyman, and other series.

The Rydberg formula is

\[ \dfrac{1}{\lambda} = R\left(\dfrac{1}{n_{1}^{2}} - \dfrac{1}{n_{2}^{2}}\right) \]

where

  • \(\lambda\) is the wavelength of the spectral line.
  • \(n_{1}\) is the lower principal index.
  • \(n_{2}\) is the upper principal index.
  • \(R\) is the Rydberg constant.

The series labels. The integers must satisfy \(n_{2} > n_{1}\). This principle is used to label each series by its lower index: Lyman \(n_{1}=1\), Balmer \(n_{1}=2\), Paschen \(n_{1}=3\).

The derivation from hydrogen levels. The same formula follows from the hydrogen energies \(E_{n}\propto -1/n^{2}\) together with \(hf=\lvert E_{i}-E_{f}\rvert\). This principle is used to derive the Rydberg formula from energy quantization.

Note: Also called the Rydberg–Balmer formula.

140.1 References

  1. Gowers, T., Barrow-Green, J., & Leader, I. (Eds.). The Princeton Companion to Mathematics. Princeton University Press, 2008. — source for the heading explanation and the definition.