Bohmian Chaos in Multinodal Bound States

We consider the Bohmian trajectories in a 2-d quantum harmonic oscillator with non commensurable frequencies whose wavefunction is of the form Ψ=aΨm1,n1(x,y)+bΨm2,n2(x,y)+cΨm3,n3(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi =a\Psi _{m_1,n_1}(x,y)+b\Psi _{m_2,n_2}(x,y)+c\Psi _{m_3,n_3}(x,y)$$\end{document}. 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m's$$\end{document}. We find (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.