View source for Einstein–Brillouin–Keller method

{{Short description|Semi-classical method for computing quantum eigenvalues}}
The '''Einstein–Brillouin–Keller''' ('''EBK''') '''method'''  is a [[Semiclassical physics|semiclassical technique]] (named after [[Albert Einstein]], [[Léon Brillouin]], and [[Joseph B. Keller]]) used to compute [[eigenvalues]] in quantum-mechanical systems.  EBK quantization is an improvement from [[Old quantum theory|Bohr-Sommerfeld quantization]] which did not consider the [[Caustic (mathematics)|caustic]] phase jumps at classical turning points.<ref>{{Cite journal |last=Einstein |first=Albert |date=1917 |title=Zum Quantensatz von Sommerfeld und Epstein |trans-title=On the Quantum Theorem of Sommerfeld and Epstein |journal=Deutsche Physikalische Gesellschaft, Verhandlungen |language=German |volume=19 |pages=82-92}}</ref><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> This procedure is able to reproduce exactly the spectrum of the 3D [[harmonic oscillator]], [[particle in a box]], and even the relativistic [[fine structure]] of the [[hydrogen]] atom.<ref>{{cite journal|title=Use of the Einstein–Brillouin–Keller action quantization|author1=Curtis, L.G.  |author2=Ellis, D.G.|journal=American Journal of Physics|year=2004|volume=72|issue=12 |doi=10.1119/1.1768554|bibcode=2004AmJPh..72.1521C|pages=1521–1523}}</ref>

In 1976–1977, [[Michael Berry (physicist)|Michael Berry]] and M. Tabor derived an extension to [[Martin Gutzwiller|Gutzwiller]] [[Quantum chaos|trace formula]] for the [[density of states]] of an [[integrable system]] starting from EBK quantization.<ref>{{cite journal|title=Closed orbits and the regular bound spectrum|author1=Berry, M.V.  |author2=Tabor, M.|journal=Proceedings of the Royal Society A|year=1976|volume=349|issue=1656 |doi=10.1098/rspa.1976.0062|bibcode=1976RSPSA.349..101B|pages=101–123|s2cid=122040979 }}</ref><ref>{{cite journal|title=Calculating the bound spectrum by path summation in action-angle variables|author1=Berry, M.V.  |author2=Tabor, M.|journal=Journal of Physics A|year=1977|volume=10|issue=3 |page=371 |doi=10.1088/0305-4470/10/3/009 |bibcode=1977JPhA...10..371B }}</ref>

There have been a number of recent results on computational issues related to this topic, for example, the work of [[Eric J. Heller]] and [[Emmanuel David Tannenbaum]] using a partial differential equation [[gradient descent]] approach.<ref>{{cite journal|title=Semiclassical Quantization Using Invariant Tori: A Gradient-Descent Approach|author1=Tannenbaum, E.D.  |author2=Heller, E.|journal=Journal of Physical Chemistry A|year=2001|volume=105|issue=12 |pages=2801–2813|doi=10.1021/jp004371d }}</ref>

==Procedure ==
Given a [[Hamilton–Jacobi equation|separable]] classical system defined by coordinates <math>(q_i,p_i);i\in\{1,2,\cdots,d\}</math>, in which every pair <math>(q_i,p_i)</math> describes a closed function or a periodic function in <math>q_i</math>, the EBK procedure involves quantizing the line integrals of <math>p_i</math>  over the closed orbit of <math>q_i</math>:
:<math>I_i=\frac{1}{2\pi}\oint p_i dq_i = \hbar \left(n_i+\frac{\mu_i}{4}+\frac{b_i}{2}\right) </math>
where <math>I_i</math> is the [[Action-angle coordinates|action-angle coordinate]], <math>n_i</math> is a positive integer, and <math>\mu_i</math> and <math>b_i</math> are [[Maslov index|Maslov indexes]]. <math>\mu_i</math> corresponds to the number of classical turning points in the trajectory of <math>q_i</math> ([[Dirichlet boundary condition]]), and <math>b_i</math> corresponds to the number of reflections with a hard wall ([[Neumann boundary condition]]).<ref>{{cite book|title=Semiclassical Physics|author1=Brack, M.  |author2=Bhaduri, R.K.|year=1997|publisher=Adison-Weasly Publishing}}</ref>

== Examples ==

=== 1D Harmonic oscillator ===
The Hamiltonian of a simple harmonic oscillator is given by
:<math>H=\frac{p^2}{2m}+\frac{m\omega^2x^2}{2}</math>
where <math>p</math> is the linear momentum and <math>x</math> the position coordinate. The action variable is given by
:<math>I=\frac{2}{\pi}\int_0^{x_0}\sqrt{2mE-m^2\omega^2x^2}\mathrm{d}x</math>
where we have used that <math>H=E</math> is the energy and that the closed trajectory is 4 times the trajectory from 0 to the turning point <math>x_0=\sqrt{2E/m\omega^2}</math>.

The integral turns out to be 
:<math>E=I\omega</math>,
which under EBK quantization there are two soft turning points in each orbit <math>\mu_x=2</math> and <math>b_x=0</math>. Finally, that yields
:<math>E=\hbar\omega(n+1/2)</math>,
which is the exact result for quantization of the quantum harmonic oscillator.

=== 2D hydrogen atom ===
The Hamiltonian for a non-relativistic electron (electric charge <math>e</math>) in a hydrogen atom is:
:<math>H=\frac{p_r^2}{2m}+\frac{p_\varphi^2}{2mr^2}-\frac{e^2}{4\pi\epsilon_{0} r}</math>
where <math>p_r</math> is the canonical momentum to the radial distance <math>r</math>, and <math>p_\varphi</math> is the canonical momentum of the azimuthal angle <math>\varphi</math>.
Take the action-angle coordinates:
:<math>I_\varphi=\text{constant}=|L|</math>
For the radial coordinate <math>r</math>:
:<math>p_r=\sqrt{2mE-\frac{L^2}{r^2}+\frac{e^2}{4\pi\epsilon_0 r}}</math>
:<math>I_r=\frac{1}{\pi}\int_{r_1}^{r_2} p_r dr = \frac{me^2}{4\pi\epsilon_0\sqrt{-2mE}}-|L| </math>
where we are integrating between the two classical turning points <math>r_1,r_2</math>  (<math>\mu_r=2</math>)
:<math>E=-\frac{me^4}{32\pi^2\epsilon_0^2(I_r+I_\varphi)^2}</math>
Using EBK quantization <math>b_r=\mu_\varphi=b_\varphi=0,n_\varphi=m</math> :
:<math> I_\varphi=\hbar m\quad;\quad m=0,1,2,\cdots </math>
:<math>I_r=\hbar(n_r+1/2)\quad;\quad n_r=0,1,2,\cdots </math>
:<math>E=-\frac{me^4}{32\pi^2\epsilon_0^2\hbar^2(n_r+m+1/2)^2}</math>
and by making <math>n=n_r+m+1</math> the spectrum of the 2D hydrogen atom <ref>{{cite book|title=Theory of Optical Processes in Semiconductors: Bulk and Microstructures|author1=Basu, P.K. |year=1997|publisher=Oxford University Press}}</ref> is recovered :
:<math>E_n=-\frac{me^4}{32\pi^2\epsilon_0^2\hbar^2(n-1/2)^2}\quad;\quad n=1,2,3,\cdots</math>
Note that for this case <math>I_\varphi=|L| </math> almost coincides with the usual quantization of the [[angular momentum operator]] on the plane <math>L_z</math>. For the 3D case, the EBK method for the total angular momentum is equivalent to the [[Langer correction]].

==See also==
{{Portal|Physics}}
*[[Hamilton–Jacobi equation]]
*[[WKB approximation]]
*[[Quantum chaos]]

==References==
* {{Cite book |last1=Duncan |first1=Anthony |title=Constructing quantum mechanics |last2=Janssen |first2=Michel |date=2019 |publisher=Oxford University Press |isbn=978-0-19-884547-8 |edition=First |location=Oxford, United Kingdom ; New York, NY |chapter=5. Guiding Principles}}
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[[Category:Quantum mechanics]]