Editing Bohr model (section)

== Refinements ==
[[Image:Sommerfeld ellipses.svg|thumb|Elliptical orbits with the same energy and quantized angular momentum]]
{{Main article|Bohr–Sommerfeld model}}

Several enhancements to the Bohr model were proposed, most notably the [[Old quantum theory|Sommerfeld or Bohr–Sommerfeld models]], which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's circular orbits.<ref name="Akhlesh Lakhtakia Ed. 1996" /> This model supplemented the quantized angular momentum condition of the Bohr model with an additional radial quantization condition, the [[William Wilson (English academic)|Wilson]]–[[Arnold Sommerfeld|Sommerfeld]] quantization condition<ref>{{Cite journal |last=A. Sommerfeld |year=1916 |title=Zur Quantentheorie der Spektrallinien |url=https://zenodo.org/record/1424309 |journal=[[Annalen der Physik]] |language=de |volume=51 |issue=17 |pages=1–94 |bibcode=1916AnP...356....1S |doi=10.1002/andp.19163561702}}</ref><ref>{{Cite journal |last=W. Wilson |year=1915 |title=The quantum theory of radiation and line spectra |url=https://zenodo.org/record/1430790 |journal=[[Philosophical Magazine]] |volume=29 |issue=174 |pages=795–802 |doi=10.1080/14786440608635362}}</ref>
: <math>\int_0^T p_r \, dq_r = n h,</math>
where ''p<sub>r</sub>'' is the radial momentum canonically conjugate to the coordinate ''q<sub>r</sub>'', which is the radial position, and ''T'' is one full orbital period. The integral is the [[Action (physics)|action]] of [[action-angle coordinates]]. This condition, suggested by the [[correspondence principle]], is the only one possible, since the quantum numbers are [[adiabatic invariant]]s.

The Bohr–Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The [[magnetic quantum number]] measured the tilt of the orbital plane relative to the ''xy''&nbsp;plane, and it could only take a few discrete values. This contradicted the obvious fact that an atom could have any orientation relative to the coordinates, without restriction. The Sommerfeld quantization can be performed in different canonical coordinates and sometimes gives different answers. The incorporation of radiation corrections was difficult, because it required finding action-angle coordinates for a combined radiation/atom system, which is difficult when the radiation is allowed to escape. The whole theory did not extend to non-integrable motions, which meant that many systems could not be treated even in principle. 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]]. The current picture of the hydrogen atom is based on the [[atomic orbitals]] of [[Schrödinger equation|wave mechanics]], which [[Erwin Schrödinger]] developed in 1926.

However, this is not to say that the Bohr–Sommerfeld model was without its successes. 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 [[Perturbation theory|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. The prevailing theory behind this difference lies in the shapes of the orbitals of the electrons, which vary according to the energy state of the electron.

The Bohr–Sommerfeld quantization conditions lead to questions in modern mathematics. Consistent semiclassical quantization condition requires a certain type of structure on the phase space, which places topological limitations on the types of symplectic manifolds which can be quantized. In particular, the symplectic form should be the [[curvature form]] of a [[Connection (mathematics)|connection]] of a [[Charles Hermite|Hermitian]] [[line bundle]], which is called a [[Geometric quantization|prequantization]].

Bohr also updated his model in 1922, assuming that certain numbers of electrons (for example, 2, 8, and 18) correspond to stable "[[Electron configuration|closed shells]]".<ref name="Shaviv">{{Cite book |last=Shaviv |first=Glora |title=The Life of Stars: The Controversial Inception and Emergence of the Theory of Stellar Structure |publisher=Springer |year=2010 |isbn=978-3642020872 |pages=203}}</ref>