Influence of a Moving Nodal Point on the "Causal Trajectories" in a Quantum Harmonic Oscillator Potential
Chaotic motion in the vicinity of a moving quantum nodal point is studied in the framework of the de Broglie–Bohm trajectory method of quantum mechanics. Chaos emerges from the sequential interaction between the quantum path with the moving nodal point depending on the distance and the frequencies between the quantum particles and their initial positions . Here, chaotic motion means the exponential divergence of initially neighboring trajectories.[more]
In a very special case (constant phase shift parameter: , ), the orbit of the nodal point is a circle with radius , with center at the origin. In most cases, the orbits of the nodal point are elliptical for different constant phase shifts. In the causal interpretation of quantum theory, the dynamics is strongly influenced by the initial distribution of the particles and the "quantum force" transmitted by the quantum potential. In this description, chaos arises because of the dynamics of the singularity of the quantum potential. At the nodal point, the quantum potential becomes very negative or approaches negative infinity, which keeps the particles from entering or passing through the nodal region. This could be interpreted as the effect that empty space, where the squared wavefunction is approximately zero, influences the motion of quantum particles via the quantum potential. The nodal point itself acts as an attractor or repeller. The motions of the quantum particles could be periodic, ergodic, or chaotic depending on the constant . There are some curves starting at the nodal point that form outward spirals . If , there are no stable limit cycles for the paths of the quantum particles, as seen in the figure.
In conclusion, moving nodal points or nodal lines are important for the appearance of chaos in the de Broglie–Bohm interpretation. This model could serve as another reference for the simplest chaotic causal trajectories . The graphic shows the squared wavefunction or the quantum potential, the vector field of the velocities (gray arrows), the trajectories of the quantum particles (colored paths), and the local minima/nodal point (blue point). The orbit of the nodal point is displayed by a thick blue line.[less]
This Demonstration studies a normalized superposition of ground state and a degenerate first excited state with a constant relative phase shift of an uncoupled isotropic harmonic oscillator in two dimensions. We assume commensurable frequencies, that is, a minimum common multiple period exists. The squared wavefunction and therefore the nodal point have period . The coefficient from the wavefunction governs the diameter of the orbit of the nodal point. The wavefunction obeys the Schrödinger equation:
with , , and so on. The normalized wavefunction for this Demonstration is:
with the eigenfunctions , where are the Hermite polynomials, are arbitrary constants (here:), are constant phase shifts, and the quantum numbers with . The velocity field is calculated from the gradient of the phase from the total wavefunction in the eikonal form . For this Demonstration the velocity is:
The quantum potential is:
In the program, if PlotPoints, AccuracyGoal, PrecisionGoal, and MaxSteps are increased, the results will be more accurate.
 C. Efthymiopoulos, C. Kalapotharakos, and G. Contopoulos, "Origin of Chaos near Critical Points of the Quantum Flow," Physical Review E 79, 2009 pp. 036203-1–036203-18. doi:10.1103/PhysRevE.79.036203, arXiv: 0903.2655 [quant-ph].
 A. J. Makowski and M. Frackowiak, "The Simplest Non-trivial Model of Chaotic Causal Dynamics," Acta Physica Polonica B 32, 2001 pp. 2831–2842. arXiv: 0111155v1 [quant-ph].