---
title: The Bohr Model of Hydrogen
module: Early Atomic Models and the Old Quantum Theory
moduleNumber: 1
lessonNumber: 2
order: 102
summary: >
  Bohr grafted three quantum postulates onto Rutherford's nuclear atom: certain
  orbits do not radiate, radiation accompanies a jump between them, and
  quantization must match classical physics for large orbits. Quantizing the
  angular momentum fixes the orbit radii and energies, reproduces the
  Rydberg-Ritz formula, and predicts the Rydberg constant from fundamental
  constants alone.
topics: [Early Atomic Models and the Old Quantum Theory]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 4 — The Nuclear Atom; §4-3 The Bohr Model of the Hydrogen Atom"
  - book: Tipler & Llewellyn
    ref: "§4-3 Reduced Mass Correction, Correspondence Principle, Fine Structure Constant, Giant Atoms"
  - book: Tipler & Mosca
    ref: "Ch. 36 — Atoms; §36-2 The Bohr Model of the Hydrogen Atom"
draft: false
---

Rutherford's nuclear atom leaves two facts unexplained. A classical electron
orbiting a nucleus is an accelerating charge, so it must radiate, spiral inward,
and reach the nucleus in under a microsecond; atoms plainly do not collapse.
And when atoms do radiate, they emit sharp lines, not the continuous sweep of
frequencies a spiraling electron would produce. In 1913 Niels Bohr resolved both
by adding three postulates to the nuclear atom. The model is wrong about the
details of electron motion, later replaced by the
[Schrödinger equation](/quantum-mechanics/wave-mechanics-1d/the-schrodinger-equation-in-one-dimension),
but its frequency condition and its energy levels are exactly right for
hydrogen, and its correspondence principle survives into modern quantum theory.

## The classical instability

Bohr took the nuclear atom for granted: an electron of charge $-e$ orbits a
nucleus of charge $+Ze$, held by the Coulomb attraction, which supplies the
centripetal force for a circular orbit of radius $r$ and speed $v$,

$$
F = \frac{kZe^2}{r^2} = \frac{mv^2}{r}.
$$

This is mechanically stable but electrically fatal. The electron accelerates
toward the center, and an accelerating charge radiates at the frequency of its
motion. As energy drains away the orbit shrinks, the frequency climbs
continuously, and the electron spirals into the nucleus. Classical physics
predicts a broadband chirp ending in collapse, the opposite of the sharp,
stable spectra observed.[^tl-classical]

$$
% caption: Left, the classical electron radiates continuously and spirals into
% the nucleus. Right, in the Bohr model the electron circles in a stationary
% state without radiating and emits a photon only when it jumps inward.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
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\fill[acc] (0,0) circle (2.6pt);
\draw[black!70, thick] (2.2,0) .. controls (2.0,1.3) and (0.8,1.7) .. (-0.6,1.0)
  .. controls (-1.6,0.4) and (-1.2,-1.0) .. (0.1,-1.1)
  .. controls (1.0,-1.1) and (1.15,-0.4) .. (0.75,0.05)
  .. controls (0.4,0.4) and (-0.2,0.25) .. (0,0);
\node[below, align=center] at (0,-1.85) {classical spiral\\(radiates, collapses)};
\begin{scope}[xshift=6.0cm]
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\node[acc, font=\scriptsize, anchor=west] at (1.65,0.7) {photon};
\node[below, align=center] at (0,-1.95) {Bohr stationary states\\(radiates only on a jump)};
\end{scope}
\end{tikzpicture}
$$

## The three postulates

Bohr's postulates are the price of admission and the whole content of the
model.[^tl-postulates]

> **Postulate (Stationary states).** The electron can occupy only certain
> discrete orbits, called stationary states, in which it does not radiate,
> despite its centripetal acceleration.

> **Postulate (Frequency condition).** The atom radiates only when the electron
> jumps from an initial state of energy $E_i$ to a final state of energy $E_f$,
> emitting a photon of frequency
> $$
> hf = E_i - E_f.
> $$
> The photon frequency is set by the energy _difference_, not by the orbital
> frequency in either state.

> **Postulate (Correspondence principle).** For large orbits and large energies,
> where the levels are closely spaced, quantum predictions must reduce to the
> classical ones.

The frequency condition is energy conservation with photon emission, and it
already contains the Ritz combination principle: writing $E_n$ for the allowed
energies, every line frequency $f = (E_i - E_f)/h$ is a difference of two members
of the term set. The remaining task is to find the $E_n$.

## Quantizing the angular momentum

In his first paper Bohr found that his results required the electron's orbital
angular momentum to be an integer multiple of $\hbar = h/2\pi$. Taking that as
the quantization rule:[^tl-quantize]

> **Definition (Angular-momentum quantization).** The orbital angular momentum
> of the electron takes only the values
> $$
> L = mvr = n\hbar, \qquad n = 1, 2, 3, \dots,
> $$
> where the integer $n$ is the principal quantum number.

Solving the force balance for the speed gives $v = (kZe^2/mr)^{1/2}$. Substituting
into $mvr = n\hbar$ and squaring eliminates $v$, leaving the quantized radii:

$$
r_n = \frac{n^2\hbar^2}{mkZe^2} = \frac{n^2}{Z}\,a_0,
\qquad
a_0 = \frac{\hbar^2}{mke^2} = 0.0529\,\text{nm}.
$$

The length $a_0$ is the **Bohr radius**, the radius of the smallest hydrogen
orbit ($n=1$, $Z=1$). Orbit radii grow as $n^2$ and shrink as $1/Z$, so
single-electron ions with $Z>1$ are more tightly bound and smaller than
hydrogen.

$$
% caption: Bohr orbits for n = 1, 2, 3 in hydrogen. Radii grow as the square of
% n, so successive orbits are spaced a0, 4a0, 9a0 from the nucleus; a photon
% carries off the energy when the electron drops to a smaller orbit.
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\fill[black!70] (0.4,0) circle (1.8pt);
\fill[black!70] (1.6,0) circle (1.8pt);
\fill[black!70] (3.4,0) circle (1.8pt);
\node[font=\scriptsize, anchor=south] at (0.42,0.05) {$a_0$};
\node[font=\scriptsize, anchor=south west] at (1.15,1.15) {$4a_0$};
\node[font=\scriptsize, anchor=south west] at (2.5,2.35) {$9a_0$};
\draw[->, acc, very thick] (-1.55,0.35) -- (-0.42,0.1);
\node[acc, font=\scriptsize, anchor=east] at (-1.55,0.55) {photon};
\end{tikzpicture}
$$

## The energy levels

Combining the force balance $\tfrac{1}{2}mv^2 = kZe^2/2r$ with the total energy
$E = \tfrac{1}{2}mv^2 - kZe^2/r$ gives $E = -kZe^2/2r$; the total energy is
negative (the electron is bound) and equal in magnitude to the kinetic energy.
Inserting the quantized radius $r_n$:[^tl-energy]

> **Definition (Bohr energy levels).** The allowed energies of a hydrogen-like
> atom are
> $$
> E_n = -\frac{mk^2Z^2e^4}{2\hbar^2}\frac{1}{n^2} = -E_0\frac{Z^2}{n^2},
> \qquad E_0 = \frac{mk^2e^4}{2\hbar^2} = 13.6\,\text{eV}.
> $$
> The lowest level $E_1 = -13.6\,\text{eV}$ (for hydrogen) is the **ground
> state**; its magnitude is the **ionization energy**, the work needed to remove
> the electron.

The energies converge on $E=0$ from below as $n\to\infty$: infinitely many
levels bunch up just under the ionization limit. This is the level structure
that the correspondence principle refers to when it speaks of closely spaced
states at large $n$.

$$
% caption: Hydrogen energy levels En = 13.6/n^2 eV below the ionization limit.
% Levels crowd toward the top as n grows; downward jumps to n = 1, 2, 3 form the
% Lyman, Balmer, and Paschen series.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% levels: y = 4.5*(1 + En/13.6)
\draw[black, thick] (0,0) -- (6.2,0);
\node[left, font=\scriptsize] at (0,0) {$n=1$};
\draw[black, thick] (0,3.375) -- (6.2,3.375);
\node[left, font=\scriptsize] at (0,3.375) {$n=2$};
\draw[black, thick] (0,4.0) -- (6.2,4.0);
\node[left, font=\scriptsize] at (0,4.0) {$n=3$};
\draw[black, thick] (0,4.22) -- (6.2,4.22);
\draw[black, thick] (0,4.32) -- (6.2,4.32);
\draw[black, dashed] (0,4.5) -- (6.2,4.5);
\node[left, font=\scriptsize, black] at (0,4.5) {ionization};
% Lyman (to n=1)
\draw[->, acc, thick] (0.7,3.375) -- (0.7,0.05);
\draw[->, acc, thick] (1.3,4.0) -- (1.3,0.05);
\draw[->, acc, thick] (1.9,4.22) -- (1.9,0.05);
\node[acc, font=\scriptsize, anchor=north] at (1.3,-0.1) {Lyman};
% Balmer (to n=2)
\draw[->, black, thick, dashed] (3.1,4.0) -- (3.1,3.43);
\draw[->, black, thick, dashed] (3.6,4.22) -- (3.6,3.43);
\draw[->, black, thick, dashed] (4.1,4.32) -- (4.1,3.43);
\node[black, font=\scriptsize, anchor=north] at (3.6,3.3) {Balmer};
% Paschen (to n=3)
\draw[->, black, thick, densely dotted] (5.3,4.22) -- (5.3,4.05);
\draw[->, black, thick, densely dotted] (5.7,4.32) -- (5.7,4.05);
\node[black, font=\scriptsize, anchor=north] at (5.5,3.85) {Paschen};
\end{tikzpicture}
$$

## The Rydberg constant from first principles

The frequency condition turns the level formula into the Rydberg-Ritz formula.
For a transition $n_i \to n_f$,

$$
hf = E_{n_i} - E_{n_f}
= E_0 Z^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right).
$$

Writing $f = c/\lambda$ and dividing by $hc$ reproduces the empirical
formula[^tl-rydberg]

$$
\frac{1}{\lambda} = Z^2 R\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right),
\qquad
R = \frac{E_0}{hc} = \frac{mk^2e^4}{4\pi c\hbar^3}.
$$

The empirical Rydberg constant, previously just a number extracted from spectra,
now emerges from $m$, $e$, $k$, $c$, and $\hbar$. Bohr's computed value matched
the spectroscopic $R = 1.097 \times 10^7\,\text{m}^{-1}$ within the uncertainty
of the constants. This is the model's decisive success: it does not fit the
spectrum, it derives it.

> **Example (The $H_\beta$ line).** The second Balmer line comes from
> $n_i = 4 \to n_f = 2$. Using the energy levels,
> $$
> E_4 = -\frac{13.6}{16} = -0.85\,\text{eV},\qquad
> E_2 = -\frac{13.6}{4} = -3.40\,\text{eV},
> $$
> so $hf = E_4 - E_2 = 2.55\,\text{eV}$ and
> $$
> \lambda = \frac{hc}{2.55\,\text{eV}} = 487\,\text{nm},
> $$
> the blue-green Balmer line. The direct Rydberg calculation
> $1/\lambda = R(\tfrac14 - \tfrac1{16})$ gives the same $486\,\text{nm}$.

## The reduced-mass correction

Bohr assumed the nucleus fixed, equivalent to giving it infinite mass. A real
nucleus of mass $M$ recoils, and both particles orbit their common center of
mass. Conservation of momentum makes the kinetic energy that of a single body of
**reduced mass**[^tl-reduced]

$$
\mu = \frac{mM}{m+M} = \frac{m}{1 + m/M}.
$$

Replacing the electron mass $m$ by $\mu$ everywhere gives the corrected Rydberg
constant

$$
R = R_\infty\,\frac{1}{1 + m/M},
$$

where $R_\infty$ uses the electron mass. The shift is about $1$ part in $2000$
for hydrogen and smaller for heavier nuclei, but it accounts for the observed
element-to-element variation of the Rydberg constant.

$$
% caption: The nucleus is not fixed; electron and nucleus both orbit their common
% center of mass, so the dynamics are those of one body of reduced mass mu about
% the center.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, dashed] (-2.9,0) circle (0.55);
\draw[black, dashed] (0.35,0) circle (2.7);
\fill[acc] (-2.9,0) circle (3.6pt);
\node[acc, font=\scriptsize, below] at (-2.9,-0.15) {nucleus};
\fill[black!70] (0.35,2.7) circle (2pt);
\node[black!70, font=\scriptsize, right] at (0.5,2.7) {electron};
\fill[black] (-2.35,0) circle (1.4pt);
\node[font=\scriptsize, below] at (-2.35,-0.15) {center of mass};
\end{tikzpicture}
$$

The correction discovered a new element. In 1931 Urey used the reduced-mass shift
of the Balmer lines to detect a second form of hydrogen with twice the mass,
**deuterium**; the two forms have the same $Z$ but different mass and are called
isotopes.

## The fine-structure constant

For the first Bohr orbit ($n=1$, $Z=1$), $mvr_1 = \hbar$ with
$r_1 = a_0 = \hbar^2/mke^2$ gives $v = ke^2/\hbar$, so the electron's speed as a
fraction of $c$ is a pure combination of constants:[^tl-fine]

> **Definition (Fine-structure constant).** The dimensionless ratio
> $$
> \alpha = \frac{ke^2}{\hbar c}
> = \frac{1.44\,\text{eV}\cdot\text{nm}}{197.3\,\text{eV}\cdot\text{nm}}
> \approx \frac{1}{137}
> $$
> is the speed of the ground-state electron in units of $c$, and it sets the
> scale of relativistic corrections to the Bohr levels.

Because $\alpha$ is built from universal constants, all observers measure the
same value. It lets the Bohr results be written compactly: the ground-state
speed is $\alpha c$, the energies scale as $\alpha^2$, and the Bohr radius is
$a_0 = \hbar/(mc\alpha)$. Sommerfeld introduced $\alpha$ while trying to explain
the observed **fine structure**, the small splitting of hydrogen lines. His
relativistic-orbit calculation gave splittings of order $\alpha^2$, matching
experiment, though the true origin is electron
[spin](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession), not orbital
eccentricity.

## The correspondence principle at work

The correspondence principle is a genuine constraint, not a slogan. For a jump
between adjacent levels $n_i = n$ and $n_f = n-1$ at large $n$, the Bohr
frequency is

$$
f = \frac{Z^2mk^2e^4}{4\pi\hbar^3}\frac{2n-1}{n^2(n-1)^2}
\;\xrightarrow{\;n\to\infty\;}\;
\frac{Z^2mk^2e^4}{2\pi\hbar^3 n^3}.
$$

The classical orbital frequency $f_\text{rev} = v/2\pi r$, evaluated with the
quantized $v$ and $r$, gives exactly

$$
f_\text{rev} = \frac{mk^2Z^2e^4}{2\pi\hbar^3 n^3},
$$

the same expression.[^tl-correspond] At large $n$ the quantum jump radiates at
the classical orbital frequency, as required. The graph of the level energies
shows why: the spacing $E_n - E_{n-1}$ shrinks toward zero as $n$ grows, so the
discrete spectrum blends into the classical continuum.

$$
% caption: Bohr energies En = 13.6/n^2 eV plotted against n. The magnitude falls
% off as the inverse square of n, and the levels crowd toward the ionization
% limit at zero, where quantum spacing becomes classically negligible.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (7.3,0) node[right, font=\scriptsize] {$n$};
\draw[->, black] (0,-4.4) -- (0,0.6) node[above, font=\scriptsize] {energy};
\draw[black, dashed] (0,0) -- (7.0,0);
\node[black, font=\scriptsize, anchor=south east] at (7.0,0.02) {ionization limit};
% points at y = -13.6/n^2 scaled by 4.0/13.6 = 0.294; x = n
\fill[acc] (1,-4.0) circle (2.4pt);
\fill[acc] (2,-1.0) circle (2.4pt);
\fill[acc] (3,-0.44) circle (2.4pt);
\fill[acc] (4,-0.25) circle (2.4pt);
\fill[acc] (5,-0.16) circle (2.4pt);
\fill[acc] (6,-0.11) circle (2.4pt);
\draw[acc, thick] (1,-4.0) .. controls (1.6,-1.9) and (2.4,-0.75) .. (3,-0.44)
  .. controls (4.0,-0.2) and (5.2,-0.12) .. (6,-0.11);
\node[acc, font=\scriptsize, anchor=north] at (1,-4.1) {ground state};
\end{tikzpicture}
$$

## Giant atoms

Since $r_n \propto n^2$, an electron nudged to very large $n$ orbits at enormous
radius. **Rydberg atoms** with $n$ in the hundreds have been made with tunable
lasers; at $n = 1000$ a hydrogen atom would be about $0.1\,\text{mm}$ across.
They are fragile because the level spacing near the ionization limit is tiny —
about $10^{-5}\,\text{eV}$ near $n=1000$, far below thermal energies of
$0.025\,\text{eV}$ — so a random collision ionizes them at once.[^tl-giant]

The Bohr model works only for one electron. Extending it to helium or any
multielectron atom fails outright, and it cannot explain the intensities or the
fine structure of the lines it does place. The
[Franck-Hertz experiment](/atomic-physics/early-models-and-old-quantum-theory/x-ray-spectra-franck-hertz)
confirms the discrete levels by a direct electrical measurement, and the
model's limits point to the wave mechanics of the
[matter-waves module](/quantum-mechanics/matter-waves/de-broglie-waves-and-electron-diffraction),
where the quantization $L = n\hbar$ reappears as the condition for a standing
electron wave to close on itself.

[^tl-classical]: **Tipler & Llewellyn**, _Modern Physics_, §4-3, Eqs. 4-12 to 4-14 — the Coulomb force balance and the classical radiative collapse of the orbit.
[^tl-postulates]: **Tipler & Llewellyn**, §4-3, Eq. 4-15 — Bohr's stationary states, the frequency condition, and the correspondence principle.
[^tl-quantize]: **Tipler & Llewellyn**, §4-3, Eqs. 4-16 to 4-19 — angular-momentum quantization, the quantized radii, and the Bohr radius.
[^tl-energy]: **Tipler & Llewellyn**, §4-3, Eqs. 4-20, 4-24 — the Bohr energy levels, ground state, and ionization energy.
[^tl-rydberg]: **Tipler & Llewellyn**, §4-3, Eqs. 4-21 to 4-23 — the frequency condition reproducing the Rydberg-Ritz formula and the derived Rydberg constant; Example 4-6.
[^tl-reduced]: **Tipler & Llewellyn**, §4-3, Eqs. 4-25 to 4-27 — the reduced-mass correction, the element-dependent Rydberg constant, and the discovery of deuterium; Example 4-7.
[^tl-fine]: **Tipler & Llewellyn**, §4-3, Eqs. 4-30 to 4-33 — the fine-structure constant, the ground-state speed, and the compact form of the Bohr results.
[^tl-correspond]: **Tipler & Llewellyn**, §4-3, Eqs. 4-28, 4-29 — the correspondence-principle demonstration that the large-n transition frequency equals the classical orbital frequency.
[^tl-giant]: **Tipler & Llewellyn**, §4-3, "Giant Atoms" — Rydberg atoms, their size scaling, and their fragility near the ionization limit.