Past exam of the theoretical physics II course of the University of Cambridge / 2012 / Paper 1 / 4 / a / Solution

The half-line square-root potential well is confining for $a>0$, which is assumed here. The hard wall imposes $\psi (0)=0$ and the smooth turning point is $x_{\mathrm{t}}=(E/a{)}^2$. With $u=a\sqrt{x}/E$, the WKB action becomesThe hard-wall and soft-turning-point WKB quantization condition therefore givesThe coefficient $a$ here denotes the potential strength, not the integration limit in the generic quantization formula.

The WKB approximation is reliable in the interior where $|k^{\prime }|/k^2\ll 1$, but fails near the soft turning point where $k\to 0$; a connection formula supplies its phase. The hard wall also requires its boundary condition, and $V^{\prime }=a/(2\sqrt{x})$ makes the local slow-variation condition fail sufficiently close to zero. Higher-energy states have a larger action relative to $\hslash$, with smaller relative endpoint regions, so the leading WKB spectrum is generally more reliable at large $n$ than for the ground state.