Quantum bouncer ($E_n=F\ell z_n,\ell =({\hslash }^2/(2mF){)}^{1/3}$)

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.

A horizontal neutron-reflecting mirror supplies the lower boundary of a quantum bouncer, with the Earth's approximately linear gravitational potential above it. An absorber above the mirror removes neutrons with sufficient vertical extent. Scanning the mirror-absorber gap and counting transmitted neutrons probes the lowest allowed gravitational states. The 2002 work led by Valery V. Nesvizhevsky reported this confinement; a subsequent detailed report describes suppression below the lowest-state height.^[ref][ref]

This is a count-versus-gap experiment, not a direct image of a wavefunction. Surface scattering, absorption and the incident vertical-motion distribution enter a quantitative transmission model.