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

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.

For $b>0$, the exponential variational trial for a square-root half-line well satisfies the wall boundary condition and has norm ${\int }_0^{\infty }{x}^2{e}^{-2bx}dx=1/(4b^3)$. integration by parts has no boundary term and givesThe supplied gamma integral givesSetting $E^{\prime }(b)=0$ yields the unique minimum,At the minimum $⟨V⟩=4⟨T⟩$, so the variational ground-state estimate isThe energy diverges as $b\to 0$ or $b\to \infty$, confirming that the stationary point minimizes the trial energy.

Write $E_{\ast }=({\hslash }^2{a}^4/(2m){)}^{1/5}$. Then $E_0^{\mathrm{WKB}}=1.811694E_{\ast }$ and $E_{\mathrm{var}}=1.876493E_{\ast }$. The variational method guarantees $E_0\le E_{\mathrm{var}}$, since the trial satisfies the wall condition and has finite energy. WKB gives no comparable upper-bound guarantee and is least controlled for the lowest state. The two estimates differ by about $3.6\%$ of the WKB value; their closeness is encouraging but does not prove that the exact energy lies between them.