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 collection 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 inappropriately 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 theory did not extend to chaotic motions, because it required a full multiply periodic trajectory of the classical system for all time in order to pose the quantum conditions.
Read more about Old Quantum Theory: Basic Principles, De Broglie Waves, Kramers Transition Matrix, Limitations of The Old Quantum Theory, History
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