Successes and Failures of Bohr

What the Bohr model got right

Bohr's 1913 model of hydrogen rests on two postulates: (1) the electron orbits in discrete circles where the angular momentum is an integer multiple of $\hbar$ ,

$L = n\hbar, \quad n = 1,2,3,\ldots,$

and (2) radiation is emitted or absorbed only when the electron jumps between two such orbits. From classical electrostatics and these two postulates alone, one can derive the allowed orbital radii and energies.

Orbital radii. Balancing the Coulomb attraction against the centripetal acceleration gives $m_e v^2 / r = e^2 / (4\pi\epsilon_0 r^2)$ . Combining with $m_e v r = n\hbar$ yields

$r_n = n^2 a_0, \quad a_0 = \frac{4\pi\epsilon_0\hbar^2}{m_e e^2} \approx 0.529\,\text{\AA},$

where $a_0$ is the Bohr radius. The ground-state radius ( $n=1$ ) matches the experimentally measured size of the hydrogen atom.

Energy levels. The total mechanical energy (kinetic plus Coulomb potential) at radius $r_n$ is

$E_n = -\frac{e^2}{8\pi\epsilon_0 a_0}\,\frac{1}{n^2} = -\frac{13.6\,\text{eV}}{n^2}.$

The ground state ( $n=1$ ) energy is $-13.6\,\text{eV}$ , so $13.6\,\text{eV}$ of energy is required to ionize hydrogen from its ground state — exactly the measured first ionization energy.

The Rydberg formula. A photon emitted when the electron falls from orbit $n_i$ to $n_f$ carries energy $h\nu = E_{n_i} - E_{n_f}$ , giving the wavenumber

$\tilde\nu = \frac{1}{\lambda} = R_\infty\!\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right),$

with $R_\infty = m_e e^4 / (8\epsilon_0^2 h^3 c) \approx 1.097 \times 10^7\,\text{m}^{-1}$ . This reproduces the Balmer series ( $n_f = 2$ , visible light), the Lyman series ( $n_f = 1$ , ultraviolet), and the Paschen series ( $n_f = 3$ , infrared) to within the experimental precision available in 1913 — a remarkable triumph.

Where the model breaks down

Despite its hydrogen success, the Bohr model is a fragile construction. Its failures are at least as instructive as its successes.

Angular momentum is wrong. The Bohr model assigns the ground state angular momentum $L = \hbar$ (the $n=1$ orbit). Full quantum mechanics shows the ground state of hydrogen has $L = 0$ — it is an $s$ state with no orbital angular momentum at all. The Bohr model gets the quantum number $n$ right but the associated angular-momentum value wrong.

Multi-electron atoms. The moment a second electron is added (helium), the Bohr scheme has no prescription for how two electrons share orbits. Every attempt to apply Bohr-style quantization to helium gives energies that disagree with experiment. Helium's ground-state energy is $-79.0\,\text{eV}$ ; a naive two-electron Bohr estimate gives roughly $-108\,\text{eV}$ . The failure compounds for heavier atoms: the model has nothing to say about the periodic table beyond hydrogen.

Spectral line intensities and selection rules. The Bohr model predicts which photon energies can be emitted but says nothing about how bright each line is. Experiments show that not all transitions are equally probable, and some are nearly forbidden. Full quantum mechanics explains this through electric-dipole selection rules ( $\Delta l = \pm 1$ , $\Delta m_l = 0,\pm 1$ ) derived from overlap integrals — concepts that do not exist in the Bohr picture.

Fine structure and spin. The hydrogen lines are not perfectly sharp; each splits into closely spaced components. This fine structure arises from relativistic corrections to the kinetic energy and from spin-orbit coupling — the interaction between the electron's magnetic moment (due to its spin) and the magnetic field it experiences as it moves through the nuclear electric field. The Bohr model has no spin and no relativistic corrections.

The anomalous Zeeman effect. When hydrogen is placed in a magnetic field, spectral lines split further (the Zeeman effect). The pattern can be "normal" or "anomalous" depending on the state. Bohr's model accounts for a simplified version of the normal Zeeman effect through orbital magnetic moments, but fails completely for the anomalous pattern, which requires electron spin — something not introduced until Goudsmit and Uhlenbeck in 1925.

No wave-particle duality or uncertainty. The model treats the electron as a classical particle moving on a well-defined orbit. It therefore violates the Heisenberg uncertainty principle: a circular Bohr orbit fixes both the radial position and radial momentum to zero simultaneously (no radial uncertainty), which is forbidden. The model is internally inconsistent with the quantum mechanics it helped inspire.

Molecules and chemical bonding. The model cannot explain why two hydrogen atoms form $\text{H}_2$ , or predict any molecular bond lengths or energies. This required the fully quantum treatment by Heitler and London in 1927.

The path forward

The Bohr model occupies a peculiar historical position: it is wrong in almost every detail except the one that matters most for hydrogen — the energy levels. Its quantization conditions are better understood as a special case of the de Broglie standing-wave condition $n\lambda = 2\pi r$ , which in turn is a forerunner of the Schrödinger equation.

The modern replacement — wave mechanics / the Schrödinger equation with the hydrogen Coulomb potential — reproduces all the Bohr energy levels exactly, and additionally gives the correct angular momentum quantum numbers, the correct wavefunctions, the selection rules, and the starting point for perturbative corrections (fine structure, Lamb shift). The lesson of Bohr is that a partially-correct model guided by the right experimental data can still reveal deep structure, even when its underlying picture is mistaken.