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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?

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  • $\begingroup$ All the online references to the EBK method that I could find are very vague, and they all seem to suggest that they are actually discussing the WKB method with another name. Is it possible that these methods are the same? $\endgroup$ Dec 17, 2016 at 13:45

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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.

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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.

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