The Balmer formula describes the wavelengths of spectral lines in the hydrogen atom.In 1885, 60-year-old Johann Jakob Balmer (a Swiss mathematician and teacher) noticed numerical harmony in the wavelengths of the hydrogen spectrum. He empirically derived a formula that was later generalized by Johannes Rydberg for all series (Lyman, Paschen, etc.). Balmer's discovery became the empirical foundation for subsequent quantum theories of atomic structure—Niels Bohr in 1913 explained the Balmer formula by postulating quantization of angular momentum.
Johann Balmer published 'Notiz über die Spektrallinien des Wasserstoffs' (Note on the Spectral Lines of Hydrogen) in 1885. His formula amazed contemporaries with its predictive accuracy. In 1888, Johannes Rydberg generalized it to other series. Niels Bohr in 1913 derived the formula from his postulates, giving it physical meaning. Arnold Sommerfeld later refined the model by introducing elliptical orbits and the fine-structure constant.
It is applied in astrophysics to determine the redshift of distant objects and analyze stellar chemical composition. Limitations: it does not account for fine or hyperfine structure; for heavy multi-electron elements the formula is inapplicable—there quantum electrodynamics is needed.
Thanks to this formula, one can tell from a star’s spectrum how much hydrogen it contains and how fast it moves.
Balmer was so fascinated by numerical patterns that he tried to explain the layout of Egyptian pyramids and even the proportions of the human body using simple fractions.
| F(Hα) | variable | flux of H-alpha line — spectral line flux watt per metre squared |
| F(Hβ) | variable | flux of H-beta line — spectral line flux watt per metre squared |
| x | variable | x-coordinate of a point on the line — x coordinate metre |
| y | variable | y-coordinate of a point on the line — y coordinate metre |
| \phi | variable | angle of the normal to the line with respect to the x-axis — angle radian |
| p | variable | perpendicular distance from the origin to the line — distance metre |
| \cos | operator | computes the cosine of an angle — cosine |
| \sin | operator | computes the sine of an angle — sine |
| f_{\text{beat}} | variable | Beat frequency — frequency hertz |
| f_1 | variable | First frequency — frequency hertz |
| f_2 | variable | Second frequency — frequency hertz |
| | \cdot | | operator | Absolute value — absolute value |




