Simple Chaotic Motion of Quantum Particles According to the Causal Interpretation of Quantum Theory

The causal interpretation of quantum theory developed by David Bohm introduced trajectories that are guided by a real phase function from the wavefunction in the polar form. A simple model is used to get chaotic motion; the trajectories undergo a transition from order to chaos depending on the relative phase factors and . The trajectories circle close to the minimum (nodal) points. For example, at the origin the squared wavefunction has a minimum for all : . The graphic shows the squared time-dependent wavefunction, the quantum particles, and the complete paths.

Chaotic motion takes place when a relative phase is introduced for a superposition of two stationary eigenstates of an isotropic harmonic quantum oscillator. The Schrödinger equation, with , is . The unnormalized wavefunction used is

, with the eigenfunctions and the energy , with .

The polar form of the wavefunction is.

The velocities and therefore the trajectories are derived from the gradient of the phase function .