Bohr–Sommerfeld model

This condition, suggested by the correspondence principle, is the only one possible, since the quantum numbers are adiabatic invariants.

In 1913, Niels Bohr displayed rudiments of the later defined correspondence principle and used it to formulate a model of the hydrogen atom which explained its line spectrum.

In the next few years Arnold Sommerfeld extended the quantum rule to arbitrary integrable systems making use of the principle of adiabatic invariance of the quantum numbers introduced by Hendrik Lorentz and Albert Einstein.

Sommerfeld made a crucial contribution[5] by quantizing the z-component of the angular momentum, which in the old quantum era was called "space quantization" (German: Richtungsquantelung).

This allowed the orbits of the electron to be ellipses instead of circles, and introduced the concept of quantum degeneracy.

Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's.

[7] The Sommerfeld model predicted that the magnetic moment of an atom measured along an axis will only take on discrete values, a result which seems to contradict rotational invariance but which was confirmed by the Stern–Gerlach experiment.

It also described the possibility of atomic energy levels being split by a magnetic field (called the Zeeman effect).

Walther Kossel worked with Bohr and Sommerfeld on the Bohr–Sommerfeld model of the atom introducing two electrons in the first shell and eight in the second.

In the end, the model was replaced by the modern quantum-mechanical treatment of the hydrogen atom, which was first given by Wolfgang Pauli in 1925, using Heisenberg's matrix mechanics.

Calculations based on the Bohr–Sommerfeld model were able to accurately explain a number of more complex atomic spectral effects.

For example, up to first-order perturbations, the Bohr model and quantum mechanics make the same predictions for the spectral line splitting in the Stark effect.

At higher-order perturbations, however, the Bohr model and quantum mechanics differ, and measurements of the Stark effect under high field strengths helped confirm the correctness of quantum mechanics over the Bohr model.