= Half-line square-root potential well
{title2=$V(x)=a\sqrt x,\quad x>0,\quad\psi(0)=0$}
A confining <Quantum Hamiltonian> on $x>0$ with $a>0$, potential $a\sqrt x$ and a <Dirichlet boundary condition> at zero has natural length $\ell=(\hbar^2/(2ma))^{2/5}$ and <energy> $E_*=(\hbar^2a^4/(2m))^{1/5}$. Rescaling $x=\ell y$ gives $H/E_*=-d^2/dy^2+\sqrt y$, so every <energy eigenvalue> scales as $E_*$. The potential <derivative> diverges at the wall even though the potential itself is finite there.
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