Bohr-Sommerfeld quantization condition from the WKB approximation How can one prove the Bohr-Sommerfeld quantization condition
$$ \oint p~dq ~=~2\pi n \hbar $$
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
$$ \frac{\partial S}{\partial t}+ \frac{(\nabla S)^{2} }{2m}+V(x)~=~0 .$$
 A: The semiclassical/Bohr-Sommerfeld/Wilson-Sommerfeld/WKB/EBK quantization rule and connection formulas are discussed in numerous textbooks. The discrete quantization condition follows from requiring single-valuedness 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. 1-7 below. For a rigorous treatment, see this Phys.SE post.
References:

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*D. Griffiths, Intro to QM, 2nd ed, 2004; chapter 9.


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


*A. Galindo & P. Pascual, QM II, 1991; chapter 9.


*P. Cvitanovic et. al., Chaos: Classical and Quantum, 2013; sections 37.1-36.7. Gutzwiller trace formula is discussed in chapter 39. The pdf file is available at www.chaosbook.org.
(Since the book is continuously updated the chapter number may shift in the future.)


*H.S. Friedrich, Theoretical Atomic Physics, 1998; section 1.5.3.


*R.G. Littlejohn, The WKB Method, lecture notes, 2019.


*Weyl law.
