Hard-wall and soft-turning-point WKB quantization (${\int }_0^{x_{\mathrm{t}}}k(x)dx=(n+3/4)\pi$)

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.

For the half-line square-root potential well, the dimensionless action integral isHence $E_n/E_{\ast }=[15\pi (n+3/4)/8{]}^{2/5}$. The ground-state estimate is $1.811694E_{\ast }$. This is a semiclassical estimate, not a variational upper bound.