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. 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. 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. 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.

Procedure

Given a separable classical system defined by coordinates, in which every pair (q_i,p_i) describes a closed function or a periodic function in q_i, the EBK procedure involves quantizing the line integrals of p_i over the closed orbit of q_i: where I_i is the action-angle coordinate, n_i is a positive integer, and \mu_i and b_i are Maslov indexes. \mu_i corresponds to the number of classical turning points in the trajectory of q_i (Dirichlet boundary condition), and b_i corresponds to the number of reflections with a hard wall (Neumann boundary condition).

Examples

1D Harmonic oscillator

The Hamiltonian of a simple harmonic oscillator is given by where p is the linear momentum and x the position coordinate. The action variable is given by where we have used that H=E 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 \mu_x=2 and b_x=0. 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 e) in a hydrogen atom is: where p_r is the canonical momentum to the radial distance r, and p_\varphi is the canonical momentum of the azimuthal angle \varphi. Take the action-angle coordinates: For the radial coordinate r: where we are integrating between the two classical turning points r_1,r_2 (\mu_r=2) Using EBK quantization : and by making n=n_r+m+1 the spectrum of the 2D hydrogen atom is recovered : Note that for this case almost coincides with the usual quantization of the angular momentum operator on the plane L_z. For the 3D case, the EBK method for the total angular momentum is equivalent to the Langer correction.