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Hydrogen transition wavelengths in the Bohr model 2026-10-18

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} .

Thus Johannes Robert Rydberg's Rydberg formula is a difference of two energy levels expressed as a spectroscopic wavenumber. The same constants determine every line; there is no new parameter for each electronic transition. With the chosen inputs,

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

produces the Balmer series. As

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.

Figure 1.
Calculated hydrogen atom energy levels and selected Lyman series and Balmer series emissions
. Original Bohr model prediction using the stipulated finite nuclear mass. Left: the full energy scale with the ionization energy threshold at zero. Right: an expanded upper-level view makes the Balmer series arrows readable. Downward arrows mark internal energy differences, not spatial Electron trajectories; their selection does not predict relative line strengths. Details.

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.

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Johann Jakob Balmer 2026-10-18

A mathematician.^[ref]

Figure 1.
Portrait of Johann Jakob Balmer
.

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