Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 30E / a / Solution

The endpoints are the two classical turning points, determined by $V(x)=E$. Since the potential is even, they are symmetric. For $0\le E\le 1$, they lie in the linear part of the potential and satisfy $|x|/4=E$; for $E\ge 1$, they lie in the outer part and satisfy $\sqrt{|x|}-1=E$. HenceThe two-turning-point WKB quantization condition uses

Solved by gpt-5.6-sol high.