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 Incoming links: Johann Jakob Balmer

For emission take integers

n_{i} > n_{f} \geq 1

. Applying conservation of energy to the model's internal energy difference, while neglecting the much smaller whole-atom photon recoil, gives

Δ E = E_{n_{i}} - E_{n_{f}} = B_{H} (\frac{1}{n _{f}^{2}} - \frac{1}{n _{i}^{2}}), B_{H} = \frac{μ e ^{4}}{2 ( 4 π ϵ _{0} ) ^{2} ℏ ^{2}},

ν = \frac{Δ E}{h}, λ = \frac{c}{ν}, \frac{1}{λ} = R_{H} (\frac{1}{n _{f}^{2}} - \frac{1}{n _{i}^{2}}), R_{H} = \frac{B _{H}}{h c} .

R_{H} = 1.096775834 \times 1 0^{7} m^{- 1}

. All wavelengths below are calculated in vacuum, not air, and are rounded model predictions rather than historical measurements.

Electronic transition $n_{i} \to n_{f}$	Series	Photon energy in electronvolts	Vacuum wavelength in $nm$	$2 \to 1$	Lyman series	$10.19872$	$121.568$	$3 \to 1$	Lyman series	$12.08737$	$102.573$	$3 \to 2$	Balmer series	$1.88865$	$656.470$	$4 \to 2$	Balmer series	$2.54968$	$486.274$	$5 \to 2$	Balmer series	$2.85564$	$434.173$	$6 \to 2$	Balmer series	$3.02184$	$410.294$

The conventional names commemorate Theodore Lyman and Johann Jakob Balmer. Fixing

n_{f} = 1

produces the ultraviolet Lyman series, beginning with the Lyman-alpha line; fixing

n_{f} = 2

n_{i} \to \infty

, the wavelength approaches

n_{f}^{2} / R_{H}

91.1763 nm

at the Lyman limit and

364.705 nm

for the Balmer series. Crowding follows directly from

E_{n + 1} - E_{n} \sim 2 B_{H} / n^{3}

. Discreteness therefore coexists with a series limit.

Correct line positions do not give wave intensities, lifetimes or electric-dipole selection rules for hydrogen. Those require a theory of the wavefunctions and their coupling to electromagnetic radiation. The circular model also omits spin, relativistic corrections and the Lamb shift. Its successful gross energy levels do not make its assumed orbits modern atomic orbitals.

Johann Jakob Balmer Created 2026-10-18 Updated 2026-11-05

A Swiss mathematician and teacher remembered for finding an empirical pattern in the hydrogen emission spectrum. In 1885 he expressed the wavelengths of its visible spectral lines through a simple formula in integer parameters. The resulting Balmer series connected measured wavelengths that had previously looked like separate experimental numbers.

Balmer's result was a numerical regularity, rather than a physical explanation of atomic structure. Its success included a further predicted line consistent with observation. The later Bohr model explained why transitions between discrete energy levels produce such a series. The episode shows how a mathematical pattern can become evidence guiding a later physical theory. ^[ref]