# 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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## 1 Answer

The semiclassical/WKB quantization rule is 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 and 2 below. For a rigorous treatment, see this Phys.SE post.

References:

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

2. H.S. Friedrich, Theoretical Atomic Physics, 1998, Section 1.5.3.

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