 Hard-wall and soft-turning-point WKB quantization

A Dirichlet boundary condition at a hard endpoint and a single smooth classical turning point give the leading WKB quantization condition ${\int }_0^{x_{\mathrm{t}}}k(x)dx=(n+3/4)\pi$, with $n=0,1,\dots$. The phase differs from the two-soft-turning-point condition because the hard wall imposes an exact wavefunction zero rather than an Airy function connection phase. The WKB validity condition is still checked in the interior.