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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The dynamic behavior for such a system is quite complex. Some of the curves are closed and periodic, while others are quasi-periodic. In the region of nodal points of the wavefunction, the trajectories apparently become accelerated and chaotic. The parameters have to be chosen carefully, because of the singularities in the velocities and the resulting large oscillations, which can lead to very unstable trajectories. The motion originates from the relative phase of the total wavefunction, which has no analog in classical particle mechanics. Further investigation to capture the full dynamics of the system is necessary.

The graphics show three-dimensional contour plots of the squared wavefunction (if enabled) and two initially neighboring trajectories. Black points mark the initial positions of the two quantum particles and green points the actual positions. Blue points indicate the nodal point structure.

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Contributed by: Klaus von Bloh (July 2015)
Open content licensed under CC BY-NC-SA


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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:

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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:

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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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