# Bohm Trajectories for a Particle in a Two-Dimensional Calogero-Moser Potential

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This Demonstration considers the trajectory of a quantum particle in a two-dimensional configuration space, in which the particle's motion in the plane is constrained by a "Calogero–Moser potential" [3, 4]. The particle can then exhibit a rich dynamical structure. In the de Broglie–Bohm (or causal) interpretation of quantum mechanics [1, 2], the particle position and momentum are well defined, and the motion can be described by continuous evolution according to the time-dependent Schrödinger equation. Chaos emerges from the sequential interaction between the quantum trajectory with the moving nodal points, depending on the distance and the frequencies between the quantum particles and their initial positions. Nodal points are created or annihilated by the singularities of the quantum amplitude . The superposition factor and the constant phase shift govern the dynamical behavior.

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Contributed by:Klaus von Bloh (March 2016)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

An entangled, unnormalized wavefunction for a two-dimensional particle, from which the trajectory is calculated, can be defined 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 , and with , , , and so on.

The motion of the particle is inextricably linked with the structure of its environment through the quantum potential with the quantum amplitude (absolute value) . Any change in the experimental setup affects the trajectory. Therefore, the trajectory cannot be measured directly.

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

References

[1] "Bohmian-Mechanics.net." (Mar 28, 2016) www.bohmian-mechanics.net/index.html.

[2] S. Goldstein. "Bohmian Mechanics." *The Stanford Encyclopedia of Philosophy*. (Mar 28, 2016)plato.stanford.edu/entries/qm-bohm.

[3] F. Calogero, "Solution of the One-Dimensional -Body problems with Quadratic and/or Inversely Quadratic Pair Potentials," *Journal of Mathematical Physics*, 12(3), 1971 pp. 419–436. doi:10.1063/1.1665604.

[4] J. Moser, "Three Integrable Hamiltonian Systems Connected with Isospectral Deformations," *Advances in Mathematics*, 16(2), 1975 pp. 197–220. doi:10.1016/0001-8708(75)90151-6.

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