Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 325 / 3 / b / Solution

For a neutron in the Earth's Newtonian gravitational potential ${\Phi }_E$, the Schrodinger equation isNear the surface, ${\Phi }_E(z)=gz$ up to an additive constant.

The Colella–Overhauser–Werner experiment uses crystal slabs as coherent beam splitters and mirrors in a neutron interferometer. The two paths contain horizontal segments of length $s$ separated vertically by $r$, then recombine. The gravitational potential energy difference is $m_{n}gr$. A neutron of speed $v$ spends time $s/v$ on a horizontal segment, so the gravitationally induced relative quantum phase isUsing the de Broglie wavelength $\lambda =h/(m_{n}v)$ and $\hslash =h/(2\pi )$ givesRotating the apparatus changes the vertical projection of its enclosed area from zero to $rs$. The number of full interference oscillations is thereforeHere $rs=(\sqrt{10}\mathrm{cm}{)}^2=10^{-3}{\mathrm{m}}^2$, soThe expected change is therefore $\boxed{9}$ oscillations to the nearest integer.