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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? $\endgroup$ – AccidentalFourierTransform Dec 17 '16 at 13:45
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I guess you mean WKB quantisation$^{[1]}$. Sommerfeld's rule precedes WBK approximation by almost a decade.

Sommerfeld's quantisation was an heuristic rule which seemed to work very well by the time, but it was known to have many limitations. It was soon superseded by Schrödinger's theory, which is much more complete.

After Schrödinger's equation was formulated, Wentzel, Kramers and Brillouin proved that Sommerfeld's rule is actually just a particular case of a more broad framework of approximation techniques that can be applied to eigenvalue problems. In other words, WKB generalised Sommerfeld's rule within the context of Schrödinger's equation. They proved that the intuitive approach by Sommerfeld was actually contained in Schrödinger's theory.

In a nutshell: Sommerfeld developed his rule prior to Schrödinger's equation, and WKB proved that the latter contains the former as a subcase. Moreover, they extended the analysis to more complex problems. This means that Sommerfeld's method and the WKB method are (essentially) equivalent, but have a very different historical origin.


$^{[1]}$ It seems that it is also called EBK quantisation, e.g., here. As far as I know, the standard name for the method is WKB, though I don't know the historical details about who was the first to develop the method.

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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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