Chaotic trajectories in complex Bohmian systems

TL;DR

This paper investigates chaotic trajectories in a 2D Bohmian quantum harmonic oscillator with multiple wavefunction components, analyzing nodal points and demonstrating how chaos arises near these nodes in complex systems.

Contribution

It provides a detailed analysis of nodal points and their influence on chaos in Bohmian trajectories for systems with multiple wavefunction components.

Findings

Nodal points can be fixed or moving, with explicit analytical forms for small quantum numbers.

Chaos is generated near nodal points where trajectories collide or approach.

Multiple scattered nodes lead to complex, chaotic particle trajectories.

Abstract

We consider the Bohmian trajectories in a 2-d quantum harmonic oscillator with non commensurable frequencies whose wavefunction is of the form $\Psi=a\Psi_{m_1,n_1}(x,y)+b\Psi_{m_2,n_2}(x,y)+c\Psi_{m_3,n_3}(x,y)$. We first find the trajectories of the nodal points for different combinations of the quantum numbers $m,n$. Then we study, in detail, a case with relatively large quantum numbers and two equal $m's$. We find %We find first the nodal points where $\Psi=0$. The nodes can be found analytically only if $m$ and $n$ are small. If two $m's$ (or two $n's$ are equal we can find explicitly the nodal points , which are of two types (1) fixed nodes independent of time and (2) moving nodes which from time to time collide with the fixed nodes and at particular times they go to infinity. Finally, we study the trajectories of quantum particles close to the nodal points and observe, for the first time, how chaos is generated in a complex system with multiple nodes scattered on the configuration space.