WKB spectrum of a square-root half-line well
= WKB spectrum of a square-root half-line well
{c}
{title2=$E_n\propto(n+3/4)^{2/5}$}
For the <half-line square-root potential well>, the <dimensionless> <action integral> is
$$
\int_0^{(E/a)^2}\frac{\sqrt{2m(E-a\sqrt x)}}{\hbar}\,dx
=\frac{8\sqrt{2m}E^{5/2}}{15\hbar a^2}.
$$
Hence $E_n/E_*=[15\pi(n+3/4)/8]^{2/5}$. The ground-state estimate is $1.811694E_*$. This is a semiclassical estimate, not a variational upper bound.