The Bohr model: Some history and context

This page is intended to give a little context behind the "Bohr model" that is introduced in the first Quantum Mechanics sheet.

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Background

Like all atomic models, the Bohr model attempts to answer the question "Why do atoms exhibit the properties that they do?", proposing a mechanical or mathematical description that can produces calculable predictions that are to be compared against experiment. Presented in 1913 by Niels Bohr and Ernest Rutherford, this model sought to reconcile a few recent observations in atomic physics:

• Atoms were known to be able to absorb and emit light, but the spectrum of the emitted light consisted of discrete wavelengths. The values of these wavelengths are described experimentally by the Rydberg formula (1888). For hydrogen, this formula states:
  $$\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) $$
  where ( n_1 ) and ( n_2 ) are integers and ( R_H ) is a constant. Rydberg had no knowledge of electrons (discovered by J. J. Thomson in 1897) and gave no physical basis for this experimental law.
• Max Planck (1900), from his work on black-body radiation, had postulated that electromatgnetic radiation was conveyed in discrete quanta of energy, called photons; the energy of a photon was related to its frequency and wavelength by
  $$ E = h f = h c / \lambda $$
  where Planck's constant ( h = 2\pi\hbar ) is an experimental constant. Planck had little justification for the idea that light was quantized, but it proved to produce accurate predictions for the spectrum of black-body radiation. Albert Einstein's 1905 work (for which he won the 1921 Nobel Prize) gave experimental confirmation of the quantization of light into photons.
• The Geiger–Marsden gold leaf experiments (1908 onwards) established that atoms consist of a heavy central nucleus of positive charge, surrounded by a diffusion of negatively charged electrons, orbiting the central charge. Rutherford gave a mathematical description (1911), although Joseph Larmor (1897) had previously proposed such a 'solar system' model.

Further back, physicists had established the laws of electromagnetism and electromagnetic radiation (Maxwell 1862), and chemists had known, at least experimentally, of the regular orbital structure elegantly illustrated in the periodic table (Mendeleev 1869).

A new model

A major shortcoming of the 'solar system' model proposed by Larmor and Rutherford was that it was incompatible with the laws of electromagnetic radiation, which predicted that an accelerating charge would emit radiation, causing the electrons to lose energy and to spiral in towards the nucleus.

Bohr's model supposes that, for unspecified reason, the electron does not lose energy from radiation. Instead, energy is allowed to change only through instantaneous absorption and emission events that move the electron from one energy level to another.

Moreover, the energy is not allowed to take any arbitrary value. Observing that the Planck constant has dimensions of angular momentum, Bohr's model requires that the angular momentum of the electron must have angular momentum equal to an integer multiple of the reduced Planck constant:

$$ m v r = n \hbar = n h / 2\pi . $$

This condition had recently been proposed by Nicholson (1912). Combined with Coulomb's law for electric attraction, the energy of the $n$th level is predicted to be (exercise):

$$ E_n = -R_E / n^2 $$

where (R_E) is a constant. This inverse square law is entirely consistent with the atomic emission spectra described by the Rydberg formula, which stated that the inverse-wavelengths of emitted photons were differences between inverse squares.

Further developments

The Bohr model successfully explains the Rydberg formula and unifies it with the Rutherford model of a nucleus; and predicts that emissions come in discrete packets of energy, although without making reference to photons. The problem, however, is how to motivate the new assumptions about the quantization of angular momentum, and to reconcile the electron acceleration with the laws of electrodynamics.

De Broglie (1924) proposed that the wave-duality duality of light might apply also to matter, proposing the formula

$$\lambda = h / p$$

for the wavelength (\lambda) for a particle with momentum (p). Experimental confirmation for the wavelike nature of electrons came from the diffraction and double-slit experiments conducted throughout the 1920s.

Under this framework, de Broglie reinterpreted the quantization condition in the Bohr model as requiring that an electron's waves be standing waves. The electron has momentum $ mv $ and executes orbits of circumference (2 \pi r), so the condition (mvr = n\hbar) asserts that the wavelength evenly divides the circumference of the orbit.