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