Maslov's method of deriving the WKB approximation

For a generic one-dimensional potential, the WKB approximation yields the quantization condition

$$\oint p dq = (n + 1/2)\hbar .$$

Here, the correction factor $1/2$ was obtained by Kramers by studying the behavior of the wave function near the turning point. It is a study in the realm of differential equation.

However, it seems that Maslov got it in quite a different way. It looks like that his approach is more topological.

So, could anyone explain Maslov's method just in the 1d setting? I know he has a book on it. But that is far beyond my comprehension.

• I have not seen this quantization condition as a consequence of the WKB approximation--or otherwise for that matter. Can you suggest where can I read up? Also, the Bohr-Sommerfeld semi-classical quantization condition is $\oint p dq = n\hbar$. Isn't the quantization condition you have written then at odds with the Bohr-Sommerfeld one? The factor of $\frac{1}{2}$ is an $\mathcal{O}(1)$ "correction" factor so I believe the two conditions cannot peacefully co-exist. Can you explain when (and why) is this condition applicable? Thanks! – Dvij Mankad Sep 20 '18 at 4:13
• Yes, the WKB condition is an improvement of the Bohr-Sommerfeld condition. For example, for the 1d harmonic oscillator, WKB gets the energies exactly, while BS gets them wrong. – Jiang-min Zhang Sep 20 '18 at 6:59