The Bohr Model of Hydrogen

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, 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 $v$,

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.¹

The three postulates

Bohr's postulates are the price of admission and the whole content of the model.²

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 $\hslash =h/2\pi$. Taking that as the quantization rule:³

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

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.

The energy levels

Combining the force balance $\frac{1}{2}m{v}^2=kZ{e}^2/2r$ with the total energy $E=\frac{1}{2}m{v}^2-kZ{e}^2/r$ gives $E=-kZ{e}^2/2r$; the total energy is negative (the electron is bound) and equal in magnitude to the kinetic energy. Inserting the quantized radius $r_n$:⁴

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 $n$.

The Rydberg constant from first principles

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

Writing $f=c/\lambda$ and dividing by $hc$ reproduces the empirical formula⁵

The empirical Rydberg constant, previously just a number extracted from spectra, now emerges from $m$, $e$, $k$, $c$, and $\hslash$. 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.

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 mass⁶

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

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.

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 ($n=1$, $Z=1$), $mv{r}_1=\hslash$ with $r_1=a_0={\hslash }^2/mk{e}^2$ gives $v=k{e}^2/\hslash$, so the electron's speed as a fraction of $c$ is a pure combination of constants:⁷

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 $a_0=\hslash /(mc\alpha )$. 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, 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

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

the same expression.⁸ 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.

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.⁹

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 confirms the discrete levels by a direct electrical measurement, and the model's limits point to the wave mechanics of the matter-waves module, where the quantization $L=n\hslash$ reappears as the condition for a standing electron wave to close on itself.