Bohr–Sommerfeld quantization

The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a set of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics. The Bohr model was the focus of study, and Arnold Sommerfeld made a crucial contribution by quantizing the z-component of the angular momentum, which in the old quantum era was called space quantization (Richtungsquantelung). This allowed the orbits of the electron to be ellipses instead of circles, and introduced the concept of quantum degeneracy. The theory would have correctly explained the Zeeman effect, except for the issue of electron spin.

The main tool was Bohr–Sommerfeld quantization, a procedure for selecting out certain discrete set of states of a classical integrable motion as allowed states. These are like the allowed orbits of the Bohr model of the atom; the system can only be in one of these states and not in any states in between.

The basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete. The system obeys classical mechanics except that not every motion is allowed, only those motions which obey the old quantum condition:

where the ${\displaystyle p_{i}}$ are the momenta of the system and the ${\displaystyle q_{i}}$ are the corresponding coordinates. The quantum numbers ${\displaystyle n_{i}}$ are integers and the integral is taken over one period of the motion at constant energy (as described by the Hamiltonian). The integral is an area in phase space, which is a quantity called the action and is quantized in units of Planck's constant. For this reason, Planck's constant was often called the quantum of action.

...
Wikipedia