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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The semiclassical/WKB quantization rule and connection formulas are discussed in numerous textbooks. The discrete quantization condition follows from requiring single-valueness of the wavefunction. Note that quantization formula gets modified by the metaplectic correction/Maslov index because of turning points.

For an elementary treatment, see e.g. Refs. 1-5 below. For a rigorous treatment, see this Phys.SE post.

References:

1. D. Griffiths, Intro to QM, 2nd ed, 2004; Chapter 9.

2. L.D. Landau & E.M. Lifshitz, QM, Vol. 3, 3rd ed, 1981; Chapter VII.

3. A. Galindo & P. Pascual, QM II, 1991; Chapter 9.

4. P. Cvitanovic et. al., Chaos: Classical and Quantum, 2013; Sections 32.1-32.3. The pdf file is available at www.chaosbook.org.

5. H.S. Friedrich, Theoretical Atomic Physics, 1998; Section 1.5.3.

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