What does this equation mean?

Above a perfectly reflecting mirror, a neutron’s Schrödinger equation is about as simple as a bound-state problem gets: an infinite wall at the mirror’s surface and a linear potential rising above it, gravity playing the same role a laser trap or a magnetic bottle plays in every other cold-atom experiment. The exact solutions are Airy functions, and the allowed energies are $E_n = \varepsilon_n \left(\frac{\hbar^2 m_{g}^2 g^2}{2 m_{i}}\right)^{1/3}$. where $\varepsilon_n$ is the n -th negative zero of the Airy function — 2.338, 4.088, 5.521, 6.787, and so on — and the mass has been split deliberately into a gravitational piece $m_{g}$ , which sets how hard the potential pulls, and an inertial piece $m_{i}$ , which sets how the wavefunction spreads in response. Nesvizhevsky and collaborators’ 2005 refinement of the original measurement, using a position-sensitive track detector to scan the standing-wave density directly above the mirror rather than only the transmitted flux, pinned down the characteristic length scale $z_0$ = 5.87 micrometres and reported the same Airy-zero ladder to several figures [ 2 ] . Feed that length scale and the lowest Airy zero into the formula above and the ground state comes out to 1.407 picoelectronvolts — a fantastically small number by nuclear or even atomic standards, corresponding to the neutron’s classical turning point at 13.7 micrometres, the hundredth-of-a-millimetre height the whole system lives at.