Rydberg formula

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The Rydberg formula relates wavelengths of hydrogenic spectral lines to pairs of principal quantum numbers. For hydrogen emission from $n_i$ to $n_f<n_i$, $1/\lambda =R_H(1/n_f^2-1/n_i^2)$, where $R_H$ includes the electron–proton reduced mass correction to the infinite-nuclear-mass Rydberg constant.

Holding the lower level fixed generates a series: $n_f=1$ gives the Lyman series and $n_f=2$ the Balmer series. Higher upper levels accumulate toward a series limit because their binding energies approach zero. For example, $3\to 2$ gives $1/\lambda =5R_H/36$, whereas the Balmer limit has $1/\lambda =R_H/4$.

The Bohr model reproduces this gross energy pattern, while a more complete treatment includes fine structure and other corrections. A precise measured line position therefore requires more specification than two principal quantum numbers alone.