Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 30A (Asymptotic Methods)

Between the classical turning points, $q<0$, while boundedness selects exponentially decaying WKB approximation solutions in the forbidden regions $x<a$ and $x>b$. The Airy turning-point connection formula at the left turning point matches the left decaying solution toin the allowed region. Matching the right decaying solution across $b$ givesFor these expressions to be proportional throughout $(a,b)$, their phases must be compatible. Each simple turning point contributes a phase shift $\pi /4$, giving the two-turning-point WKB quantization condition

Write $r=e^x$, $ϵ=\hslash /\sqrt{2m}$, $L=l+1/2$, and $\lambda =-|\lambda |$. Since $V(r)=-V_0/r$, the transformed equation hasThe two turning radii $a<b$ are the positive roots ofsoInside the allowed interval,The semiclassical action integral is thereforeEquating this with $(n+1/2)\pi$ from part (a), and using $L=l+1/2$, givesSince ${ϵ}^2={\hslash }^2/(2m)$,Thus the Coulomb WKB quantization with the Langer correction yields