Einstein–Brillouin–Keller method

The Einstein–Brillouin–Keller method (EBK) is a semiclassical method (named for 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 wich 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, Berry and 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.

Given a separable classical system defined by coordinates ${\displaystyle (q_{i},p_{i});i\in \{1,2,\cdots ,d\}}$ and every ${\displaystyle (q_{i},p_{i})}$ pair describe a closed function or a periodic function in ${\displaystyle q_{i}}$ then the EBK procedure involves quantizing the path integrals of ${\displaystyle p_{i}}$ over the closed orbit of ${\displaystyle q_{i}}$  :

...
Wikipedia