= Hydrogen transition wavelengths in the Bohr model
{title2=$\lambda^{-1}=R_H(n_f^{-2}-n_i^{-2})$}
For emission take integers $n_i>n_f\geq1$. Applying <conservation of energy> to the model's internal <energy> difference, while neglecting the much smaller whole-atom <photon> recoil, gives
$$
\Delta E=E_{n_i}-E_{n_f}
=B_H\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right),\qquad
B_H=\frac{\mu e^4}{2(4\pi\epsilon_0)^2\hbar^2},
$$
$$
\nu=\frac{\Delta E}{h},\qquad
\lambda=\frac{c}{\nu},\qquad
\frac{1}{\lambda}=R_H\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right),\qquad
R_H=\frac{B_H}{hc}.
$$
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, $R_H=1.096775834\times10^7\,\mathrm{m^{-1}}$. 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 $\mathrm{nm}$
| $2\to1$
| <Lyman series>
| $10.19872$
| $121.568$
| $3\to1$
| <Lyman series>
| $12.08737$
| $102.573$
| $3\to2$
| <Balmer series>
| $1.88865$
| $656.470$
| $4\to2$
| <Balmer series>
| $2.54968$
| $486.274$
| $5\to2$
| <Balmer series>
| $2.85564$
| $434.173$
| $6\to2$
| <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 $n_f^2/R_H$: $91.1763\,\mathrm{nm}$ at the <Lyman limit> and $364.705\,\mathrm{nm}$ for the <Balmer series>. Crowding follows directly from $E_{n+1}-E_n\sim2B_H/n^3$. Discreteness therefore coexists with a series limit.
\Image[/bohr-hydrogen-levels-and-transitions.png]
{title=Calculated <hydrogen atom> <energy levels> and selected <Lyman series> and <Balmer series> emissions}
{description=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.}
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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