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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.

A WKB quantization condition selects approximate bound-state energies by requiring an oscillatory wavefunction to match both endpoint conditions. For a smooth one-dimensional well bounded by two separated simple classical turning points $a(E),b(E)$,Each soft turning point contributes an Airy turning-point connection formula phase. A hard Dirichlet wall and one soft turning point instead give $(n+3/4)\pi$, as in hard-wall and soft-turning-point WKB quantization. Two regular Dirichlet endpoints with no turning point give an integer multiple of $\pi$. The endpoint type is part of the formula.

For the quantum harmonic oscillator, the two-turning-point action is $\pi E/\omega$. The condition gives $E_n=\hslash \omega (n+1/2)$, which happens to agree with the exact spectrum. In a general potential it is a leading semiclassical estimate, and nearby or degenerate turning points can invalidate independent connection formulas.