Einstein–Brillouin–Keller method

The Einstein–Brillouin–Keller (EBK) method is a semiclassical method (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 $$(q_i,p_i);i\in\{1,2,\cdots,d\}$$, 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 $$q_i$$:
 * $$I_i=\frac{1}{2\pi}\oint p_i dq_i = \hbar \left(n_i+\frac{\mu_i}{4}+\frac{b_i}{2}\right) $$

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).

1D Harmonic oscillator
The Hamiltonian of a simple harmonic oscillator is given by
 * $$H=\frac{p^2}{2m}+\frac{m\omega^2x^2}{2}$$

where $$p$$ is the linear momentum and $$x$$ the position coordinate. The action variable is given by
 * $$I=\frac{2}{\pi}\int_0^{x_0}\sqrt{2mE-m^2\omega^2x^2}\mathrm{d}x$$

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 $$x_0=\sqrt{2E/m\omega^2}$$.

The integral turns out to be
 * $$E=I\omega$$,

which under EBK quantization there are two soft turning points in each orbit $$\mu_x=2$$ and $$b_x=0$$. Finally, that yields
 * $$E=\hbar\omega(n+1/2)$$,

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:
 * $$H=\frac{p_r^2}{2m}+\frac{p_\varphi^2}{2mr^2}-\frac{e^2}{4\pi\epsilon_{0} r}$$

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:
 * $$I_\varphi=\text{constant}=|L|$$

For the radial coordinate $$r$$:
 * $$p_r=\sqrt{2mE-\frac{L^2}{r^2}+\frac{e^2}{4\pi\epsilon_0 r}}$$
 * $$I_r=\frac{1}{\pi}\int_{r_1}^{r_2} p_r dr = \frac{me^2}{4\pi\epsilon_0\sqrt{-2mE}}-|L| $$

where we are integrating between the two classical turning points $$r_1,r_2$$ ($$\mu_r=2$$)
 * $$E=-\frac{me^4}{32\pi^2\epsilon_0^2(I_r+I_\varphi)^2}$$

Using EBK quantization $$b_r=\mu_\varphi=b_\varphi=0,n_\varphi=m$$ :
 * $$ I_\varphi=\hbar m\quad;\quad m=0,1,2,\cdots $$
 * $$I_r=\hbar(n_r+1/2)\quad;\quad n_r=0,1,2,\cdots $$
 * $$E=-\frac{me^4}{32\pi^2\epsilon_0^2\hbar^2(n_r+m+1/2)^2}$$

and by making $$n=n_r+m+1$$ the spectrum of the 2D hydrogen atom is recovered :
 * $$E_n=-\frac{me^4}{32\pi^2\epsilon_0^2\hbar^2(n-1/2)^2}\quad;\quad n=1,2,3,\cdots$$

Note that for this case $$I_\varphi=|L| $$ 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.