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Mathematics for Physics

83 Rydberg Formula

The formula that predicts the wavelengths of light emitted or absorbed when an electron transitions between energy shells in hydrogen-like atoms that is used to compute those wavelengths from integer shell indices.

Note: Also called the Rydberg–Balmer formula.

definition [d] (Rydberg Formula) From Gowers, Barrow-Green, and Leader:

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

where \(n_{1}\) and \(n_{2}\) are integers, \(n_{1} < n_{2}\), and \(R\) is known as the Rydberg constant.

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.

83.1 Elementary Example

83.1.1 Simple

The Balmer line with \(n_{1} = 2\) and \(n_{2} = 3\) has

\[ \dfrac{1}{\lambda} = R\left(\dfrac{1}{4} - \dfrac{1}{9}\right) = R\cdot\dfrac{5}{36} \]

where

  • \(\lambda\) is the emitted wavelength.
  • \(R\) is the Rydberg constant.

83.1.2 General

Three Balmer lines use \(n_{1} = 2\) with \(n_{2} = 3,4,5\).

\[ \dfrac{1}{\lambda_{n_{2}}} = R\left(\dfrac{1}{4} - \dfrac{1}{n_{2}^{2}}\right),\quad n_{2} \in \{3,4,5\} \]

where

  • each \(\lambda_{n_{2}}\) is a Balmer wavelength.