Bohr Model

The Bohr model is an old quantum model of the hydrogen atom. It explains important spectral regularities by imposing quantized angular momentum and transition frequencies on a classical-looking orbit picture. It is historically crucial and mathematically useful, but it is not modern quantum mechanics.

Hydrogen and Atomic Stability

Before Bohr, two facts stood in tension:

atoms are stable enough to form matter,
hydrogen emits and absorbs light at sharply defined frequencies.

Classical electromagnetism makes a charged electron in orbit radiate. A naive classical atom should lose energy and collapse. Empirical spectral formulas, such as Balmer’s formula for visible hydrogen lines, showed regularity without explaining why only certain frequencies appear.

Bohr’s 1913 model joined Rutherford’s nuclear atom to quantum conditions. It did not abandon classical pictures completely; instead it restricted them by new postulates.

Bohr Postulates

In a compact modern retelling, the model assumes:

an electron can occupy certain stationary orbits without radiating continuously;
angular momentum is quantized as

L=n\hbar, \qquad n=1,2,3,\ldots;

radiation is emitted in a transition to a lower-energy stationary state, with

h\nu=E_i-E_f.

For absorption the atom gains energy instead: $h\nu=E_f-E_i>0$ . These rules were radical because they suspended the classical expectation of continuous radiation from accelerated charge.

Bohr model schematic with quantized circular orbits and energy-level transitions — The Bohr model combines a classical-looking orbit picture with old quantum rules. The modern hydrogen atom keeps the energy-level idea but replaces literal electron orbits with wavefunctions and angular-momentum eigenstates.

Energy Levels

For a circular Coulomb orbit, combine centripetal force with angular-momentum quantization. Write $k_e=1/(4\pi\epsilon_0)$ and let $\mu$ be the electron–proton reduced mass:

\frac{\mu v^2}{r}=\frac{k_e e^2}{r^2}, \qquad \mu vr=n\hbar.

Eliminating $v$ gives $r_n=n^2\hbar^2/(\mu k_e e^2)$ . Since the kinetic energy is half the magnitude of the negative potential energy,

E_n=-\frac{k_e e^2}{2r_n} =-\frac{\mu k_e^2e^4}{2\hbar^2n^2}.

The first orbit has a radius about $0.529\,\text{Å}$ . The nonrelativistic hydrogen energies are therefore approximately

E_n \simeq -\frac{13.6\,\mathrm{eV}}{n^2}, \qquad n=1,2,3,\ldots.

Spectral lines arise from differences:

h\nu = 13.6\,\mathrm{eV} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right), \qquad n_i>n_f.

This relation reproduces the Rydberg pattern for hydrogen. The agreement was a major success because it connected a measured spectral constant to atomic structure and Planck’s constant.

Successes

The model explained:

the scale of the hydrogen atom,
the Rydberg formula for hydrogen spectral lines,
a lowest allowed orbit, with stability imposed by the nonradiation postulate,
the role of Planck’s constant in atomic structure,
why radiation frequencies correspond to energy differences rather than orbital frequencies.

It also gave a concrete route from spectroscopy to quantized energy levels.

Failures

The Bohr model fails as a general theory:

it is not systematic for multi-electron atoms,
it does not explain line intensities or transition probabilities,
it relies on special orbit quantization rules,
it does not naturally include spin,
it cannot replace noncommuting observables and Hilbert-space states,
it treats electron orbits in a way modern quantum mechanics does not preserve.

The Sommerfeld Model extended old quantum theory, but it did not remove the framework’s conceptual patchwork character.

Why It Is Not Modern Quantum Mechanics

Historical caution: Bohr orbits are not literal modern electron paths. In wave mechanics, the hydrogen atom is solved by a Hamiltonian eigenvalue problem. States are wavefunctions with angular-momentum quantum numbers, probability densities, and degeneracies. The energy formula survives, but the picture that supports it changes.

The difference can be tested immediately with angular momentum. Bohr assigns the first orbit $L=\hbar$ . The modern hydrogen ground state has $\ell=0$ , hence $L^2=\hbar^2\ell(\ell+1)=0$ . Matching gross energies does not imply matching states or observables.

In the modern hydrogen solution, normalizability and boundary conditions produce the bound spectrum. The angular-momentum algebra determines the allowed orbital labels; no selected circular path is needed.

Common Mistakes

Drawing Bohr orbits as if they were modern electron trajectories.
Saying the Bohr model is wrong in a way that erases its real hydrogen-spectrum success.
Forgetting that spectral frequencies come from energy differences.
Applying the Bohr model uncritically to many-electron atoms.
Treating angular-momentum quantization in old quantum theory as the same as the modern angular-momentum operator spectrum.

Exercises

Calculate the photon energy and approximate wavelength for $n_i=3\to n_f=2$ , using $hc=1240\,\mathrm{eV\,nm}$ .

Solution

\Delta E=13.6\left(\frac14-\frac19\right)\mathrm{eV} \simeq1.89\,\mathrm{eV},\qquad \lambda=\frac{hc}{\Delta E}\simeq656\,\mathrm{nm}.

This is the red Hα line. The precision is limited by rounding the energy scale and neglecting finer corrections.

Does agreement with the hydrogen energy formula establish Bohr’s angular-momentum rule?

Solution

No. The modern ground state has zero orbital angular momentum, whereas Bohr assigns it $\hbar$ . Energy measurements alone do not specify the angular-momentum content of the states. Separate predictions must be compared separately with experiment.

References

N. Bohr, “On the Constitution of Atoms and Molecules,” Philosophical Magazine 26, 1-25 (1913), DOI: 10.1080/14786441308634955.
M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed., American Institute of Physics, 1989.
J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Springer, 1982-2001.
Nobel Prize Outreach, Niels Bohr Facts.
E. Rutherford, “The Scattering of α and β Particles by Matter and the Structure of the Atom,” Philosophical Magazine 21, 669-688 (1911), DOI: 10.1080/14786440508637080.