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The Colella–Overhauser–Werner experiment observed a gravitationally induced relative quantum phase in a neutron interferometer. The 1975 primary report identifies the measured signal as a phase shift caused by the interaction of neutron matter waves with Earth's gravitational field.^[ref]

An idealized two-path model explains the sensitivity. If equal-speed horizontal segments of length $L$ are separated vertically by $h$, their gravitational potential energy difference is $mgh$ and their common traversal time is approximately $L/v$. The corresponding phase-difference magnitude isThe sign depends on path orientation and phase convention. Recombination converts this difference into matter-wave interference, making a tiny potential-energy difference measurable through phase accumulation. The rectangular equal-speed expression is a leading explanatory model, not a complete precision correction for real crystal diffraction, bending and rotation effects.

The de Broglie wavelength is the distance over which a matter wave accumulates one full spatial quantum phase cycle in its propagation direction:Here $h$ is the Planck constant, $\mathbf{p}$ is the momentum, and $\mathbf{k}$ is the angular wavevector. The scalar $p$ in $\lambda =h/p$ means the positive momentum magnitude, not a signed component. Reversing a momentum reverses the wavevector without making the wavelength negative. At zero momentum there is no finite spatial phase period: the limiting wavelength diverges.

This relation makes microscopic diffraction accessible with particles: increasing the momentum shortens the wavelength until it is comparable with the spacings in a crystal scattering problem. The electron accelerating-voltage wavelength gives a practical way to tune this scale, while the Davisson–Germer experiment tests it by comparing electron diffraction directions with independently known crystal geometry. The same relation applies to neutral particles in a neutron interferometer; electric charge is not a requirement for a matter wave.

The wavelength belongs to the wavefunction, not to a sinusoidal path of a particle. A free-particle momentum convention makes the distinction especially clear: a plane wave has a rotating complex quantum phase but a constant probability density. A localized wave packet contains a range of momenta and therefore generally has a range of wavelengths, rather than one exact spatial period.

For a neutron in the Earth's Newtonian gravitational potential ${\Phi }_E$, the Schrödinger 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.