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

time steps

50

initialize

trajectory with density

no

yes

α

2.

x

0

initial position

0.5

y

0

initial position

0.5

z

0

initial position

0.9

initial distance

0.04

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

dx

, on the boundary of the harmonic potential.

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.