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] Given a separable classical system defined by coordinates
describes a closed function or a periodic function in
, the EBK procedure involves quantizing the line integrals of
over the closed orbit of
is the action-angle coordinate,
is a positive integer, 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] 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
Finally, that yields which is the exact result for quantization of the quantum harmonic oscillator.
The Hamiltonian for a non-relativistic electron (electric charge
) in a hydrogen atom is: where
is the canonical momentum to the radial distance
is the canonical momentum of the azimuthal angle
: where we are integrating between the two classical turning points
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.