What is the difference between Einstein–Brillouin–Keller (EBK) quantization and Bohr-Sommerfeld quantization? They look the same, right? They all give the same quantization condition
$$ \int_{C_j } p d q = 2 \pi \hbar n_j . $$
So, what is the difference between the two?
The correct EBK quantization rule is :
$$ \int_{C_j } p d q = 2 \pi \hbar \ \left( \, n_j \, + \, \frac{\gamma_j}{4} \, \right) $$
where the second integer $ \gamma_j $ is the Maslov index (or Keller-Maslov index), a topological number discovered by Joseph B. Keller in 1958 (and independantly by Victor P. Maslov). This Maslov index was absent from the old Bohr-Sommerfeld theory.
(Example : for the one-dimensional harmonic oscillator, one has : $ \gamma = 2 $, and the EBK quantization rule yields the exact quantum spectrum ; omitting the Maslov index would gives a wrong result.)
Beforehand, Einstein had remarked (already in 1917) that this kind of quantization rule could only be applied to integrable systems, whose phase space possess invariant tori.
Read e.g. : A. Douglas Stone, Einstein's unknown insight and the problem of quantizing chaos, Physics Today 58 (8) (August 2005), 37-43.
It's an improvement of WKB. In WKB, we need to know a classical trajectory and do the phase space integral. But in more than 2D, the orbit may not be closed (central force motion with potential not $r^2$ or $1/r$). So EBK improves the way to do the semi-classical quantization. They are based on the same philosophy by Bohr-Sommerfield.
