View source for Physics:Bohr–Sommerfeld model

{{Short description|Extension of the Bohr model to allow elliptical orbits of electrons around an atomic nucleus}}
{{quantum mechanics | cTopic = Background }}

[[File: Drawing_of_Sommerfeld_atom.jpg |thumb|310px|The Sommerfeld extensions of the 1913 solar system Bohr model of the hydrogen atom showing the addition of elliptical orbits to explain spectral fine structure.]]

The '''Bohr–Sommerfeld model''' (also known as the '''Sommerfeld model''' or '''Bohr–Sommerfeld theory''') was an extension of the [[Physics:Bohr model|Bohr model]] to allow elliptical orbits of electrons around an atomic nucleus.  Bohr–Sommerfeld theory is named after Danish physicist [[Biography:Niels Bohr|Niels Bohr]] and German physicist [[Biography:Arnold Sommerfeld|Arnold Sommerfeld]].  Sommerfeld argued that if electronic orbits could be [[Ellipse|elliptical]] instead of circular, the energy of the electron would be the same, except in the presence of a magnetic field, introducing what is now known as quantum degeneracy.

The Bohr–Sommerfeld model supplemented the quantized angular momentum condition of the Bohr model with an additional radial quantization condition, the [[Biography:William Wilson (English academic)|Wilson]]–[[Biography:Arnold Sommerfeld|Sommerfeld]] quantization condition<ref>
{{cite journal
 |author=A. Sommerfeld
 |year=1916
 |title=Zur Quantentheorie der Spektrallinien
 |language=de
 |journal=[[Physics:Annalen der Physik|Annalen der Physik]]
 |volume=51 |issue= 17|pages=1–94
 |bibcode=1916AnP...356....1S
 |doi=10.1002/andp.19163561702
 |url=https://zenodo.org/record/1424309
 }}</ref><ref>
{{cite journal
 |author=W. Wilson
 |year=1915
 |title=The quantum theory of radiation and line spectra
 |journal=[[Physics:Philosophical Magazine|Philosophical Magazine]]
 |volume=29 |pages=795–802
 |doi=10.1080/14786440608635362
 |issue=174
|url=https://zenodo.org/record/1430790
 }}</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'', which is the radial position, and ''T'' is one full orbital period. The integral is the [[Physics:Action|action]] of [[Physics:Action-angle coordinates|action-angle coordinates]]. This condition, suggested by the [[Physics:Correspondence principle|correspondence principle]], is the only one possible, since the quantum numbers are [[Physics:Adiabatic invariant|adiabatic invariant]]s.

== History ==

{{main article|Physics:Old quantum theory}}

In 1913, [[Biography:Niels Bohr|Niels Bohr]] displayed rudiments of the later defined [[Physics:Correspondence principle|correspondence principle]] and used it to formulate a model of the [[Physics:Hydrogen atom|hydrogen atom]] which explained the line spectrum. In the next few years Arnold Sommerfeld extended the quantum rule to arbitrary integrable systems making use of the principle of [[Physics:Adiabatic invariant|adiabatic invariance]] of the quantum numbers introduced by Lorentz and Einstein. Sommerfeld made a crucial contribution<ref name="Sommerfeld">{{cite book|first=Arnold |last= Sommerfeld|title=Atombau und Spektrallinien'|  publisher=Friedrich Vieweg und Sohn|location=Braunschweig|year=1919|isbn=978-3-87144-484-5}}</ref> by quantizing the z-component of the [[Physics:Angular momentum|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. The theory would have correctly explained the [[Physics:Zeeman effect|Zeeman effect]], except for the issue of electron [[Physics:Spin|spin]]. Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's.

In the 1950s Joseph Keller updated Bohr–Sommerfeld quantization using Einstein's interpretation  of 1917,<ref>The Collected Papers of Albert Einstein, vol. 6, A. Engel, trans., Princeton U. Press,
Princeton, NJ (1997), p. 434</ref> now known as [[Physics:Einstein–Brillouin–Keller method|Einstein–Brillouin–Keller method]]. In 1971, [[Biography:Martin Gutzwiller|Martin Gutzwiller]] took into account that this method only works for integrable systems and derived a [[Physics:Quantum chaos|semiclassical way of quantizing chaotic systems]] from [[Physics:Path integral formulation|path integrals]].<ref>{{cite journal|title=Einstein's unknown insight and the problem of quantizing chaos|author=Stone, A.D.|journal=Physics Today|date=August 2005|volume=58|number=8|pages=37–43|url=https://www.eng.yale.edu/stonegroup/publications/phys_today.pdf|doi=10.1063/1.2062917|bibcode = 2005PhT....58h..37S }}</ref>

== Predictions ==

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 [[Physics:Stern–Gerlach experiment|Stern–Gerlach experiment]].  This was a significant step in the development of quantum mechanics.  It also described the possibility of atomic energy levels being split by a [[Physics:Magnetic field|magnetic field]] (called the Zeeman effect).  [[Biography:Walther Kossel|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.<ref>{{cite journal | jstor=228422 | title=The Kossel-Sommerfeld Theory and the Ring Atom | last1=Heilbron | first1=John L. | journal=Isis | year=1967 | volume=58 | issue=4 | pages=450–485 | doi=10.1086/350299 | s2cid=144639796 }}</ref>

== Issues ==

The Bohr–Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The [[Physics:Magnetic quantum number|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 be turned this way and that 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 [[Biography:Wolfgang Pauli|Wolfgang Pauli]] in 1925, using Heisenberg's [[Physics:Matrix mechanics|matrix mechanics]]. The current picture of the hydrogen atom is based on the atomic orbitals of [[Physics:Schrödinger equation|wave mechanics]], which [[Biography:Erwin Schrödinger|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 [[Physics:Perturbation theory|perturbations]], the Bohr model and quantum mechanics make the same predictions for the spectral line splitting in the [[Physics:Stark effect|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|curvature form]] of a [[Connection (mathematics)|connection]] of a [[Biography:Charles Hermite|Hermitian]] [[Line bundle|line bundle]], which is called a [[Geometric quantization|prequantization]].

== Relativistic orbit ==

[[File:Sommerfeld ellipses.svg|thumb|Elliptical orbits with the same energy and quantized angular momentum]]


Arnold Sommerfeld derived the relativistic solution of atomic energy levels.<ref name="Sommerfeld" /> We will start this derivation<ref>https://archive.org/details/atombauundspekt00sommgoog/page/n541 - Atombau und Spektrallinien, 1921, page 520</ref> with the relativistic equation for energy in the [[Physics:Electric potential|electric potential]]

:<math>W={m_\mathrm{0} c^2} \left( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} - 1 \right) - k \frac{Z e^2}{r}</math>

After substitution <math>u = \frac{1}{r}</math> we get

:<math>\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} = 1 + \frac{W}{m_\mathrm{0} c^2} + k \frac{Z e^2}{m_\mathrm{0} c^2} u</math>

For momentum <math>p_\mathrm{r} = m \dot{r}</math>,  <math>p_\mathrm{\varphi} = m r^2 \dot{\varphi}</math> and their ratio <math>\frac{p_\mathrm{r}}{p_\mathrm{\varphi}} = - \frac{du}{d \varphi}</math> the equation of motion is (see [[Physics:Binet equation|Binet equation]])

:<math>\frac{d^2 u}{d \varphi ^2} = - \left( 1 - k^2 \frac{Z^2 e^4}{c^2 p_\mathrm{\varphi}^2} \right) u + \frac{m_\mathrm{0} kZe^2}{p_\mathrm{\varphi}^2} \left( 1+\frac{W}{m_\mathrm{0} c^2} \right) = - \omega_\mathrm{0}^2 u + K </math>

with solution

:<math> u = \frac{1}{r} = K + A \cos \omega_\mathrm{0} \varphi </math>

The angular shift of periapsis per revolution is given by

:<math> \varphi_\mathrm{s} = 2 \pi \left(\frac{1}{\omega_\mathrm{0}} - 1\right) \approx 4 \pi^3 k^2 \frac{Z^2 e^4}{c^2 n_\mathrm{\varphi}^2 h^2} </math>

With the quantum conditions

:<math> \oint p_\mathrm{\varphi} \, d \varphi = 2 \pi p_\mathrm{\varphi} = n_\mathrm{\varphi} h</math>

and

:<math> \oint p_\mathrm{r} \, d r = p_\mathrm{\varphi} \oint \left( \frac{1}{r} \frac{dr}{d \varphi} \right)^2 \, d \varphi = n_\mathrm{r} h</math>

we will obtain energies

:<math> \frac{W}{m_\mathrm{0} c^2} = \left( 1 + \frac{\alpha ^2 Z^2}{\left(n_\mathrm{r} + \sqrt{n_\mathrm{\varphi} ^2 - \alpha ^2 Z^2} \right)^2} \right) ^{-1/2} - 1</math>

where <math>\alpha</math> is the [[Physics:Fine-structure constant|fine-structure constant]]. This solution (using [[Substitution (algebra)|substitution]]s for quantum numbers) is equivalent to the solution of the [[Physics:Dirac equation|Dirac equation]].<ref name="Granovski">{{cite journal|author=Ya I Granovski |url=http://www.iop.org/EJ/article/1063-7869/47/5/L06/PHU_47_5_L06.pdf |title=Sommerfeld formula and Dirac's theory |journal=Physics-Uspekhi |volume=47 |issue=5 |pages=523&ndash;524 |year=2004|doi=10.1070/PU2004v047n05ABEH001885|bibcode = 2004PhyU...47..523G |s2cid=250900220 }}</ref> Nevertheless, both solutions fail to predict the [[Physics:Lamb shift|Lamb shift]]s.

== See also ==


* [[Physics:Bohr model|Bohr model]]
* [[Physics:Old quantum theory|Old quantum theory]]

== References ==

<references />

{{Quantum mechanics topics}}

{{DEFAULTSORT:Bohr-Sommerfeld model}}
[[Category:Atomic physics]]
[[Category:Hydrogen physics]]
[[Category:Foundational quantum physics]]
[[Category:History of physics]]
[[Category:Old quantum theory]]

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