2.1: Fine structure of spectral lines

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Recall that the energy levels of a hydrogen atom are given by

\[E_n = -{13.6\;{\rm eV} \over n^2}\;\;\;\;\;\;\;\;\;n=1,2,...,\infty\]

This will give rise to a series of spectral lines at a set of allowed transition frequencies when the electron in the atom is excited.

\[\omega_{n_i\rightarrow n_f} = {E_{n_i}-E_{n_f} \over \hbar}\]

Here, \(n_i\) and \(n_f\) represent the initial energy level to which the electron is excited and \(n_f\) represents the final level to which it decays. In the decay process, electromagnetic radiation is emitted which can be detected in a spectrometer.

If one examines, the Lyman spectrum, for example, which corresponds to \(n_f=1\) and produces spectral lines in the ultraviolet part of the electromagnetic spectrum, one finds that the individual lines are actually several lines of nearly identical frequency. For example, the \(2P \rightarrow 1s\) transition is actually a doublet, with the two components being separated by \(~10^{-4}\) eV which is also on the order of a few tens of wavenumbers, which is about \(10^5\) times smaller than the splitting predicted from the formula, i.e., 10.2 eV. Clearly, the simple theory based on the above formulae is not sufficient to explain the multiplicity of lines actually observed.