# Bohr-Sommerfeld quantization from the WKB approximation

How can one prove the Bohr-Sommerfeld quantization formula

$$\oint p~dq ~=~2\pi n \hbar$$

from the WKB ansatz solution $$\Psi(x)~=~e^{iS(x)/ \hbar}$$ for the Schroedinger equation?

With $S$ the action of the particle defined by Hamilton-Jacobi equation

$$\frac{\partial S}{\partial t}+ \frac{(\nabla S)^{2} }{2m}+V(x)~=~0 .$$

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