$$E_n = \varepsilon_n \left(\frac{\hbar^2 m_{g}^2 g^2}{2 m_{i}}\right)^{1/3}$$

Equation 1 · Crossed Fields

The Oldest Quantum-Gravity Experiment Is a Neutron Falling

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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…

Meanings in this article

• $\varepsilon_n$: the n -th negative zero of the Airy function — 2.
• $m_{i}$: the inertial piece.

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