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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In the causal approach, the quantum potential (qp) is responsible for the dynamics of the particle and it describes the detailed behavior of the system. The qp does not depend on the intensity of the wave, but rather on its form; it need not fall off with increasing distance.

The graphic shows the trajectory (white), the velocity vector field (red), the nodal points (blue), the absolute value of the wavefunction, the Calogero–Moser potential (black), and the initial and final points of the trajectory (shown as white points). With the checkbox enabled, you can see contour lines of the quantum potential (yellow). You can return to the original settings with the "initialize" checkbox.

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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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