 Quantum bouncer

A quantum particle above a hard reflecting wall in a linearly increasing potential $V(x)=Fx$ for $x>0$. The Schrödinger equation reduces to the Airy equation. Decay at infinity selects $\mathrm{Ai}$, while the wall requires $\mathrm{Ai}(-z_n)=0$. Thus ${\psi }_n\propto \mathrm{Ai}(x/\ell -z_n)$ and the energies are the displayed values. The $n$th state has $n-1$ interior nodes and a decaying tail beyond its classical turning point. A gravitational realization has $F=mg$; gravitational neutron bound-state transmission tests such confinement experimentally.