A beam of neutrons with energy E runs horizontally into a...

A beam of neutrons with energy $E$ runs horizontally into a crystal. The crystal transmits half the neutrons and deflects the other half vertically upwards. After climbing to height $H$ these neutrons are deflected through $90^{\circ}$ onto a horizontal path parallel to the originally transmitted beam. The two horizontal beams now move a distance $L$ down the laboratory, one distance $H$ above the other. After going distance $L$, the lower beam is deflected vertically upwards and is finally deflected into the path of the upper beam such that the two beams are co-spatial as they enter the detector. Given that particles in both the lower and upper beams are in states of well-defined momentum, show that the wavenumbers $k, k^{\prime}$ of the lower and upper beams are related by $$ k^{\prime} \simeq k\left(1-\frac{m_{\mathrm{n}} g H}{2 E}\right) $$ In an actual experiment (R. Colella et al., Phys. Rev. Let., $\mathbf{3 4}, 1472$, 1975) $E=0.042 \mathrm{eV}$ and $L H \sim 10^{-3} \mathrm{~m}^{2}$ (the actual geometry was slightly different). Determine the phase difference between the two beams at the detector. Sketch the intensity in the detector as a function of $H$.

A beam of neutrons with energy $E$ runs horizontally into a crystal. The crystal transmits half the neutrons and deflects the other half vertically upwards. After climbing to height $H$ these neutrons are deflected through $90^{\circ}$ onto a horizontal path parallel to the originally transmitted beam. The two horizontal beams now move a distance $L$ down the laboratory, one distance $H$ above the other. After going distance $L$, the lower beam is deflected vertically upwards and is finally deflected into the path of the upper beam such that the two beams are co-spatial as they enter the detector. Given that particles in both the lower and upper beams are in states of well-defined momentum, show that the wavenumbers $k, k^{\prime}$ of the lower and upper beams are related by
$$
k^{\prime} \simeq k\left(1-\frac{m_{\mathrm{n}} g H}{2 E}\right)
$$
In an actual experiment (R. Colella et al., Phys. Rev. Let., $\mathbf{3 4}, 1472$, 1975) $E=0.042 \mathrm{eV}$ and $L H \sim 10^{-3} \mathrm{~m}^{2}$ (the actual geometry was slightly different). Determine the phase difference between the two beams at the detector. Sketch the intensity in the detector as a function of $H$.

Key Concepts

Matter?Wave Interference

This concept underpins the idea that particles, such as neutrons, display wave-like properties when propagating. The interference pattern arises because the wave functions associated with different paths combine, leading to constructive or destructive interference. This is a fundamental aspect of quantum mechanics that demonstrates the wave-particle duality of matter.

de Broglie Wavelength and Wave Number

A particle’s de Broglie wavelength is inversely related to its momentum, with the relationship ? = h/p, where h is Planck’s constant. The wave number k, which is defined as 2?/?, plays a critical role in describing how the phase of the particle’s wave function evolves in space. Any change in momentum, such as from an external potential, alters the wavelength and wave number accordingly.

Phase Difference in Interferometry

In interferometry, the phase difference between two coherent beams is key to understanding interference patterns. This phase difference arises from differences in the accumulated phase along different paths, which can be influenced by factors like path length, potential energy variations, or changes in the particle's momentum. The resulting interference determines the detected intensity pattern.

Gravitational Potential Effects on Quantum Phases

Particles moving in a gravitational field experience a change in potential energy, which can affect their kinetic energy and, consequently, their momentum and de Broglie wavelength. This results in a gravitationally induced phase shift between particles following different trajectories. Calculations often show that the phase shift is proportional to the height difference in the gravitational field, thus affecting the interference outcome.

Interferometric Detection and Intensity Modulation

The intensity at a detector in an interferometric setup is a function of the superposition of the wave functions from different paths, and it depends critically on their relative phase difference. Small changes in the experimental parameters, such as the path height in a gravitational field, result in variations in the phase difference, which in turn modulate the observed intensity pattern in a predictable, oscillatory manner.

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