equation

Balmer formula

The Balmer formula describes the wavelengths of spectral lines in the hydrogen atom.
6 related articles · 3 formulas

Description

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.

History

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.

How it works

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.

In practice

Thanks to this formula, one can tell from a star’s spectrum how much hydrogen it contains and how fast it moves.

Fun fact

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.

Concept graph →

Formulas

$$\frac{F(H\alpha)}{F(H\beta)} \approx 2.86$$
Hydrogen line ratio under standard recombination conditions.
What is in the formula
F(Hα)variableflux of H-alpha line — spectral line flux watt per metre squared
F(Hβ)variableflux of H-beta line — spectral line flux watt per metre squared
\frac{hα}{hβ}\gg2.86 · 2606.04711v1
$$x\cos\phi + y\sin\phi = p$$
A line's equation using normal distance and angle.
What is in the formula
xvariablex-coordinate of a point on the line — x coordinate metre
yvariabley-coordinate of a point on the line — y coordinate metre
\phivariableangle of the normal to the line with respect to the x-axis — angle radian
pvariableperpendicular distance from the origin to the line — distance metre
\cosoperatorcomputes the cosine of an angle — cosine
\sinoperatorcomputes the sine of an angle — sine
π\cosφ-σ\sinφ=c · 2607.24694v1
$$f_{\text{beat}} = |f_1 - f_2|$$
Beat frequency equals the absolute difference of two frequencies.
What is in the formula
f_{\text{beat}}variableBeat frequency — frequency hertz
f_1variableFirst frequency — frequency hertz
f_2variableSecond frequency — frequency hertz
| \cdot |operatorAbsolute value — absolute value
f_{{биений}}=|f_1-f_2| · 2607.27999v1

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