# Nonlocality in the de Broglie-Bohm Interpretation of Quantum Mechanics

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One of the counterintuitive features of quantum mechanics is the phenomenon of nonlocality. In simple terms, this implies that, in some circumstances, particles that have interacted at some initial time and then become spatially separated remain "entangled", such that a measurement on one particle affects the other instantaneously, no matter how large the separation of the two particles has become.

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Contributed by:Klaus von Bloh (March 2016)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Consider the Schrödinger equation

, with , , , , and so on. The solution involves associated Laguerre polynomials. An entangled, un-normalized wavefunction for two one-dimensional particles, which cannot move along the entire and axes but are constrained to remain on the half and axes, can be expressed by a superposition state with a special parameter :

,

where , are eigenfunctions, and are permuted eigenenergies of the corresponding stationary one-dimensional Schrödinger equation, with and . The eigenfunctions , are given by

, with and ,

where , are associated Laguerre polynomials, and are the quantum numbers with . The wavefunction is taken from [1].

For , the velocity vector in one coordinate direction does not depend on the other coordinate direction:

and ,

from which the trajectory is calculated, and where and are the initial starting positions, which can be freely chosen for numerical integration of the velocity vector. The initial starting positions (particle 1) and (particle 2) can be changed by using the controls. For , , and , both components of the velocity are equal, which produces a straight line in configuration space.

For the special case , the trajectory becomes periodic, depending only on the constant phase shift in the term.

In the program, if PlotPoints, AccuracyGoal, PrecisionGoal, and MaxSteps are increased (if enabled), the results will be more accurate.

References

[1] M. Trott, *The Mathematica GuideBook for Symbolics*, New York: Springer, 2006.

[2] B.-G. Englert, M. O. Scully, G. Süssman, and H. Walther, "Surrealistic Bohm Trajectories," *Zeitschrift für Naturforschung A*, 47(12), 1992 pp. 1175–1186.

[3] "Bohmian-Mechanics.net." (Mar 16, 2016) www.bohmian-mechanics.net/index.html.

[4] S. Goldstein, "Bohmian Mechanics," *The Stanford Encyclopedia of Philosophy*. (Mar 16, 2016)plato.stanford.edu/entries/qm-bohm.

[5] S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. Krister Shalm, and A. M. Steinberg, "Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer," *Science*, 332(6034), 2011 pp. 1170–1173. doi:10.1126/science.1202218.

[6] D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg,"Experimental Nonlocal and Surreal Bohmian Trajectories,"* Science Advances,* 2(2), 2016 pp. 1–7. doi:10.1126/science.1501466.

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