Bohr model
Derivation

If an electron in an atom is moving on an orbit with period T, classically the electromagnetic radiation will repeat itself every orbital period. If the coupling to the electromagnetic field is weak, so that the orbit doesn't decay very much in one cycle, the radiation will be emitted in a pattern which repeats every period, so that the Fourier transform will have frequencies which are only multiples of 1/T. However, in the quantum realm, the quantization of angular momentum leads to discrete energy levels, and the emitted frequencies are quantized according to the energy differences between these levels. This discrete nature of energy levels introduces a fundamental departure from the classical radiation law, giving rise to distinct spectral lines in the emitted radiation.

In quantum mechanics, this emission must be in quanta of light, of frequencies consisting of integer multiples of 1/ T, so that classical mechanics is an approximate description at large quantum numbers. This means that the energy level corresponding to a classical orbit of period 1/ T must have nearby energy levels which differ in energy by h / T, and they should be equally spaced near that level,

Bohr worried whether the energy spacing 1/ T should be best calculated with the period of the energy state E n {\displaystyle E_{n}}, or E n + 1 {\displaystyle E_{n+1}}, or some average—in hindsight, this model is only the leading semiclassical approximation.

Bohr considered circular orbits. Classically, these orbits must decay to smaller circles when photons are emitted. The level spacing between circular orbits can be calculated with the correspondence formula. For a hydrogen atom, the classical orbits have a period T determined by Kepler's third law to scale as r. The energy scales as 1/ r, so the level spacing formula amounts to

It is possible to determine the energy levels by recursively stepping down orbit by orbit, but there is a shortcut.

The angular momentum L of the circular orbit scales as r {\displaystyle {\sqrt {r}}}. The energy in terms of the angular momentum is then

Assuming, with Bohr, that quantized values of L are equally spaced, the spacing between neighboring energies is

This is as desired for equally spaced angular momenta. If one kept track of the constants, the spacing would be ħ, so the angular momentum should be an integer multiple of ħ,

This is how Bohr arrived at his model.

An electron in the lowest energy level of hydrogen (n = 1) therefore has about 13.6 eV less energy than a motionless electron infinitely far from the nucleus. The next energy level (n = 2) is −3.4 eV. The third (n = 3) is −1.51 eV, and so on. For larger values of n, these are also the binding energies of a highly excited atom with one electron in a large circular orbit around the rest of the atom. The hydrogen formula also coincides with the Wallis product.

The combination of natural constants in the energy formula is called the Rydberg energy (R):

This expression is clarified by interpreting it in combinations that form more natural units:

Since this derivation is with the assumption that the nucleus is orbited by one electron, we can generalize this result by letting the nucleus have a charge q = Ze, where Z is the atomic number. This will now give us energy levels for hydrogenic (hydrogen-like) atoms, which can serve as a rough order-of-magnitude approximation of the actual energy levels. So for nuclei with Z protons, the energy levels are (to a rough approximation):

The actual energy levels cannot be solved analytically for more than one electron (see n -body problem) because the electrons are not only affected by the nucleus but also interact with each other via the Coulomb force.

When Z = 1/ α (Z ≈ 137), the motion becomes highly relativistic, and Z cancels the α in R; the orbit energy begins to be comparable to rest energy. Sufficiently large nuclei, if they were stable, would reduce their charge by creating a bound electron from the vacuum, ejecting the positron to infinity. This is the theoretical phenomenon of electromagnetic charge screening which predicts a maximum nuclear charge. Emission of such positrons has been observed in the collisions of heavy ions to create temporary super-heavy nuclei.

The Bohr formula properly uses the reduced mass of electron and proton in all situations, instead of the mass of the electron,

However, these numbers are very nearly the same, due to the much larger mass of the proton, about 1836.1 times the mass of the electron, so that the reduced mass in the system is the mass of the electron multiplied by the constant 1836.1/(1+1836.1) = 0.99946. This fact was historically important in convincing Rutherford of the importance of Bohr's model, for it explained the fact that the frequencies of lines in the spectra for singly ionized helium do not differ from those of hydrogen by a factor of exactly 4, but rather by 4 times the ratio of the reduced mass for the hydrogen vs. the helium systems, which was much closer to the experimental ratio than exactly 4.

For positronium, the formula uses the reduced mass also, but in this case, it is exactly the electron mass divided by 2. For any value of the radius, the electron and the positron are each moving at half the speed around their common center of mass, and each has only one fourth the kinetic energy. The total kinetic energy is half what it would be for a single electron moving around a heavy nucleus.
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