Editing Old quantum theory (section)

== Basic principles ==
{{See also|Jun Ishiwara#Quantum physics}}
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 ''quantization condition'':

:<math>
\oint_{H(p,q)=E} p_i \, dq_i = n_i h
</math>

where the <math>p_i</math> are the momenta of the system and the <math>q_i</math> are the corresponding coordinates. The quantum numbers <math>n_i</math> are ''integers'' and the integral is taken over one period of the motion at constant energy (as described by the [[hamiltonian mechanics|Hamiltonian]]). The integral is an area in phase space, which is a quantity called the action and is quantized in units of the (unreduced) [[Planck constant]]. For this reason, the Planck constant was often called the ''quantum of action''.

In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates <math>q_i</math> in terms of which the motion is periodic. The periods of the different motions do not have to be the same, they can even be incommensurate, but there must be a set of coordinates where the motion decomposes in a multi-periodic way.

The motivation for the old quantum condition was the [[correspondence principle]], complemented by the physical observation that the quantities which are quantized must be [[adiabatic invariant]]s. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a general system up to an additive constant.

This quantization condition is often known as the ''Wilson–Sommerfeld rule'',<ref>{{Cite book|title=Introduction to quantum mechanics : with applications to chemistry|last1=Pauling|first1=Linus|last2=Wilson|first2=Edgar Bright|publisher=Dover Publications|year=2012|isbn=9780486134932|location=New York, N.Y.|oclc=830473042|author-link=Linus Pauling|author-link2=Edgar Bright Wilson}}</ref> proposed independently by [[William Wilson (English academic)|William Wilson]]<ref>{{Cite journal|last=Wilson|first=William|author-link=William Wilson (English academic)|date=1915|title=LXXXIII. The quantum-theory of radiation and line spectra|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|volume=29|issue=174|pages=795–802|doi=10.1080/14786440608635362|url=https://zenodo.org/record/1430790}}</ref> and Arnold Sommerfeld.<ref>{{Cite journal|last=Sommerfeld|first=Arnold|author-link=Arnold Sommerfeld|date=1916|title=Zur Quantentheorie der Spektrallinien|journal=Annalen der Physik|language=en|volume=356|issue=17|pages=1–94|doi=10.1002/andp.19163561702|issn=0003-3804|bibcode=1916AnP...356....1S|url=https://zenodo.org/record/1424309}}</ref>