Chaotic Quantum Motion of Two Particles in a 3D Harmonic Oscillator Potential
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
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.
[more]
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.
Permanent Citation