# Chaotic Quantum Motion of Two Particles in a 3D Harmonic Oscillator Potential

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A system with three degrees of freedom, consisting of a superposition of three coherent stationary eigenfunctions with commensurate energy eigenvalues and a constant relative phase, can exhibit chaotic motion in the de Broglie–Bohm formulation of quantum mechanics (see Quantum Motion of Two Particles in a 3D Trigonometric Pöschl–Teller Potential). We consider here an analog using a three-dimensional harmonic-oscillator potential. In this case, the velocities of the particles are autonomous, with a complex, chaotic trajectory structure. Two particles are placed randomly, separated by an initial distance , on the boundary of the harmonic potential.

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Contributed by: Klaus von Bloh (July 2015)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Associated Hermite polynomials arise as the solution of the Schrödinger equation: , with , , and so on. A degenerate, unnormalized, complex-valued wavefunction for the three-dimensional case can be given by:

,

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

,

where , , are Hermite polynomials. The parameter is a constant phase shift. The eigenvalues' numbers depend on the three quantum numbers .

In this Demonstration, the wavefunction is defined by:

.

In this case, the square of the Schrödinger wavefunction , where is its complex conjugate, is not time dependent:

.

The velocity field is calculated from the gradient of the phase from the total wavefunction in the eikonal form (often called polar form) . The time-dependent phase function from the total wavefunction is:

.

The corresponding velocity field becomes time independent (autonomous) because of the time-independent gradient of the phase function.

In the program, if PlotPoints, AccuracyGoal, PrecisionGoal, MaxSteps, and MaxIterations are enabled, increasing them will give more accurate results.

References

[1] "Bohmian-Mechanics.net." (Jul 30, 2015) www.bohmian-mechanics.net/index.html.

[2] S. Goldstein. "Bohmian Mechanics." *The Stanford Encyclopedia of Philosophy*. (Jul 30, 2015)plato.stanford.edu/entries/qm-bohm.

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