# Collision of Two Neutrons in the de Broglie–Bohm Approach

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Neutron scattering is a spectroscopic method of measuring the motions of magnetic particles. This can be used to probe a wide variety of different physical phenomena: interference effects, diffusional or hopping motions of atoms, rotational modes of molecules, acoustic modes and molecular vibrations, recoil in quantum fluids and magnetic quantum excitations [1].

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Contributed by: Klaus von Bloh (January 2020)

Open content licensed under CC BY-NC-SA

## Details

The Schrödinger equation for the motion of a free particle is given by:

.

It is readily shown that the general normalized solution of the two sub-beams is of the form

with

where is the considered point of space and is the midpoint of the wave. The dispersion relationship holds with , here (atomic units). The solution as stated is defined in the infinite space . Obviously, the initial condition at determines (inverse Fourier transform). It follows

.

For the neutron beam, the is chosen by:

with initial density , , and with the wavenumber vector for the particle. The total wavefunction becomes

.

From the total wavefunction for , the equation for the phase function could be calculated, and therefore the components of the velocity. This is a standard procedure [5–7] in Bohmian mechanics, which will not be described here.

When PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps are increased (if enabled), the results will be more accurate. The initial distance between the starting trajectories is determined by the factor .

References

[1] Wikipedia. "Neutron Spectroscopy." (Jan 21, 2020) en.wikipedia.org/wiki/Neutron_spectroscopy.

[2] H. R. Brown, C. Dewdney and G. Horton, "Bohm Particles and Their Detection in the Light of Neutron Interferometry," *Foundations of Physics, *25(2), 1995 pp. 329–347. doi:10.1007/BF02055211.

[3] C. Dewdney, "The Quantum Potential Approach to Neutron Interferometry Experiments,"* Physica B+C*, 151(1–2), 1988 pp. 160–170. doi:10.1016/0378-4363(88)90161-1.

[4] D. M. Greenberger, "The Neutron Interferometer as a Device for Illustrating the Strange Behavior of Quantum Systems," *Reviews of Modern Physics, *55(4)*,* 1983 pp. 875-905. doi:10.1103/RevModPhys.55.875.

[5] "Bohmian-Mechanics.net." (Jan 21, 2020) www.bohmian-mechanics.net/index.html.

[6] S. Goldstein. "Bohmian Mechanics." *The Stanford Encyclopedia of Philosophy*. (Jan 14, 2020)plato.stanford.edu/entries/qm-bohm.

[7] P. R. Holland, *The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics*, New York: Cambridge University Press, 1993.

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