Bohm Trajectories for a Particle in a Two-Dimensional Calogero-Moser Potential

time steps

50

constant phase factor δ

1.5

superposition factor a

0.1

quantum potential

initialize

This Demonstration considers the trajectory of a quantum particle in a two-dimensional configuration space, in which the particle's motion in the

xy

plane is constrained by a "Calogero–Moser potential" [3, 4]. The particle can then exhibit a rich dynamical structure. In the de Broglie–Bohm (or causal) interpretation of quantum mechanics [1, 2], the particle position and momentum are well defined, and the motion can be described by continuous evolution according to the time-dependent Schrödinger equation. Chaos emerges from the sequential interaction between the quantum trajectory with the moving nodal points, depending on the distance and the frequencies between the quantum particles and their initial positions. Nodal points are created or annihilated by the singularities of the quantum amplitude

R

. The superposition factor

a

and the constant phase shift

δ

govern the dynamical behavior.

In the causal approach, the quantum potential (qp) is responsible for the dynamics of the particle and it describes the detailed behavior of the system. The qp does not depend on the intensity of the wave, but rather on its form; it need not fall off with increasing distance.

The graphic shows the trajectory (white), the velocity vector field (red), the nodal points (blue), the absolute value of the wavefunction, the Calogero–Moser potential (black), and the initial and final points of the trajectory (shown as white points). With the checkbox enabled, you can see contour lines of the quantum potential (yellow). You can return to the original settings with the "initialize" checkbox.