# Bohm Trajectories for a Special Type of a Pseudoharmonic Potential

Bohm Trajectories for a Special Type of a Pseudoharmonic Potential

This Demonstration is the three-dimensional version of "Bohm Trajectories for an Isotropic Harmonic Oscillator with Added Inverse Quadratic Potential." It considers the Schrödinger equation that connects an isotropic harmonic oscillator potential with an additional centrifugal potential containing in the de Broglie–Bohm approach. This special pseudoharmonic-type potential has the form .

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2

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V(r)=A+

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r

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i

2

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In de Broglie–Bohm theory, the particle has a well-defined trajectory in configuration space calculated from the total phase function. An appealing feature of the Bohmian description is that one can impose initial positions for the trajectories, just as in any classical dynamical system, in which the final positions of the particles are determined by their initial positions. Such initial positions are not controllable by the experimenter, so there is an appearance of randomness in the pattern of detection. The wavefunction guides the particles in such a way that the particles avoid the regions in which the wave density is small and are attracted to the regions in which the wave density is large [1, 2]. In the regions with small wave density, the particles will be accelerated. The actual point-particles moving under the influence of the wavefunction are clearly seen by the oscillations of the two orbits in the graphics, which depends, among other things, on the parameter .

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Exact solutions of the Schrödinger equation for this potential have the form of confluent hypergeometric functions. An analogous potential in three dimensions can represent the interaction of some diatomic molecules [3]. Obviously, the pseudoharmonic oscillator with =0 behaves asymptotically as a harmonic oscillator, but has a singularity at For , there is a small region where the potential exhibits a repulsive inverse-square-type character.

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r=0.

-0.125<<0

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The graphics show the wave density (if enabled), the velocity vector field (red arrows), the initial starting points of two possible orbits (red points, shown as small red spheres), the actual position (colored points, shown as small spheres) and two possible trajectories with the initial distance .

δx