Einstein–Brillouin–Keller method explained
The Einstein–Brillouin–Keller (EBK) method is a 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 Bohr-Sommerfeld quantization which did not consider the caustic phase jumps at classical turning points.[1] [2] 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.[3]
In 1976–1977, Michael Berry and M. Tabor derived an extension to Gutzwiller trace formula for the density of states of an integrable system starting from EBK quantization.[4] [5]
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.[6]
Procedure
Given a separable classical system defined by coordinates
(qi,pi);i\in\{1,2, … ,d\}
, in which every pair
describes a closed function or a periodic function in
, the EBK procedure involves quantizing the line integrals of
over the closed orbit of
:
where
is the
action-angle coordinate,
is a positive integer, and
and
are Maslov indexes.
corresponds to the number of classical turning points in the trajectory of
(
Dirichlet boundary condition), and
corresponds to the number of reflections with a hard wall (
Neumann boundary condition).
[7] Examples
1D Harmonic oscillator
The Hamiltonian of a simple harmonic oscillator is given by
where
is the linear momentum and
the position coordinate. The action variable is given by
where we have used that
is the energy and that the closed trajectory is 4 times the trajectory from 0 to the turning point
.
The integral turns out to be
,which under EBK quantization there are two soft turning points in each orbit
and
. Finally, that yields
,which is the exact result for quantization of the quantum harmonic oscillator.
2D hydrogen atom
The Hamiltonian for a non-relativistic electron (electric charge
) in a hydrogen atom is:
where
is the canonical momentum to the radial distance
, and
is the canonical momentum of the azimuthal angle
.Take the action-angle coordinates:
For the radial coordinate
:
}
prdr=
| me2 |
4\pi\epsilon0\sqrt{-2mE |
}-|L| where we are integrating between the two classical turning points
(
)
Using EBK quantization
br=\mu\varphi=b\varphi=0,n\varphi=m
:
I\varphi=\hbarm ; m=0,1,2, …
Ir=\hbar(nr+1/2) ; nr=0,1,2, …
and by making
the spectrum of the 2D hydrogen atom
[8] is recovered :
Note that for this case
almost coincides with the usual quantization of the
angular momentum operator on the plane
. For the 3D case, the EBK method for the total angular momentum is equivalent to the
Langer correction.
See also
References
- Book: Duncan . Anthony . Constructing quantum mechanics . Janssen . Michel . 2019 . Oxford University Press . 978-0-19-884547-8 . First . Oxford, United Kingdom ; New York, NY . 5. Guiding Principles.
Notes and References
- Einstein . Albert . 1917 . Zum Quantensatz von Sommerfeld und Epstein . On the Quantum Theorem of Sommerfeld and Epstein . Deutsche Physikalische Gesellschaft, Verhandlungen . German . 19 . 82-92.
- Einstein's unknown insight and the problem of quantizing chaos. Stone, A.D.. Physics Today. August 2005. 58. 8. 37–43. 10.1063/1.2062917. 2005PhT....58h..37S .
- Use of the Einstein–Brillouin–Keller action quantization. Curtis, L.G. . Ellis, D.G.. American Journal of Physics. 2004. 72. 12 . 10.1119/1.1768554. 2004AmJPh..72.1521C. 1521–1523.
- Closed orbits and the regular bound spectrum. Berry, M.V. . Tabor, M.. Proceedings of the Royal Society A. 1976. 349. 1656 . 10.1098/rspa.1976.0062. 1976RSPSA.349..101B. 101–123. 122040979 .
- Calculating the bound spectrum by path summation in action-angle variables. Berry, M.V. . Tabor, M.. Journal of Physics A. 1977. 10. 3 . 371 . 10.1088/0305-4470/10/3/009 . 1977JPhA...10..371B .
- Semiclassical Quantization Using Invariant Tori: A Gradient-Descent Approach. Tannenbaum, E.D. . Heller, E.. Journal of Physical Chemistry A. 2001. 105. 12 . 2801–2813. 10.1021/jp004371d .
- Book: Semiclassical Physics. Brack, M. . Bhaduri, R.K.. 1997. Adison-Weasly Publishing.
- Book: Theory of Optical Processes in Semiconductors: Bulk and Microstructures. Basu, P.K. . 1997. Oxford University Press.