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Physica Scripta

Chaotic Bohmian trajectories and the role of quantum potential in rational frequencies system

Umair Abdul Halim, Nurisya Mohd Shah, Kar Tim Chan, Ahmad Hazazi Ahmad Sumadi

Published Mar 25, 2025• 1 citation
Publisher

Abstract

Abstract Bohmian mechanics is an alternative formulation of quantum mechanics that maintains the notion of a well-defined trajectory governed by the wave function. The extra potential in this framework called quantum potential plays an important role in determining the dynamics of the Bohmian trajectory. In this study, we examine numerically the characteristics of a rational frequency system near a moving nodal point and how quantum potential plays an important role in generating chaotic motion for a specific mixture of wavefunction amplitudes. We show the transition from a regular to a chaotic trajectory when there is an abrupt change of quantum potential near the nodal point. This phenomenon occurs in the centre where the value of quantum potential is much higher than classical potential.

DOI
10.1088/1402-4896/adc518

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Abstract Bohmian mechanics is an alternative formulation of quantum mechanics that maintains the notion of a well-defined trajectory governed by the wave function. The extra potential in this framework called quantum potential plays an important role in determining the dynamics of the Bohmian trajectory. In this study, we examine numerically the characteristics of a rational frequency system near a moving nodal point and how quantum potential plays an important role in generating chaotic motion for a specific mixture of wavefunction amplitudes. We show the transition from a regular to a chaotic trajectory when there is an abrupt change of quantum potential near the nodal point. This phenomenon occurs in the centre where the value of quantum potential is much higher than classical potential.
We explore the transition from order to chaos for the Bohmian trajectories of a simple quantum system corresponding to the superposition of three stationary states in a 2D harmonic well with incommensurable frequencies. We study in particular the role of nodal points in the transition to chaos. Our main findings are (a) a proof of the existence of bounded domains in configuration space which are devoid of nodal points, (b) an analytical construction of formal series representing regular orbits in the central domain as well as a numerical investigation of its limits of applicability, (c) a detailed exploration of the phase-space structure near the nodal point. In this exploration we use an adiabatic approximation and we draw the flow chart in a moving frame of reference centered at the nodal point. We demonstrate the existence of a saddle point (called X-point) in the vicinity of the nodal point which plays a key role in the manifestation of exponential sensitivity of the orbits. One of the invariant manifolds of the X-point continues as a spiral terminating at the nodal point. We find cases of Hopf bifurcation at the nodal point and explore the associated phase space structure of the nodal point—X-point complex. We finally demonstrate the mechanism by which this complex generates chaos. Numerical examples of this mechanism are given for particular chaotic orbits, and a comparison is made with previous related works in the literature.
Bohmian mechanics is a causal interpretation of quantum mechanics in which particles describe trajectories guided by the wave function. The dynamics in the vicinity of nodes of the wave function, usually called vortices, is regular if they are at rest. However, vortices generically move during time evolution of the system. We show that this movement is the origin of chaotic behavior of quantum trajectories. As an example, our general result is illustrated numerically in the two-dimensional isotropic harmonic oscillator.
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)Ψ=aΨ_{m_1,n_1}(x,y)+bΨ_{m_2,n_2}(x,y)+cΨ_{m_3,n_3}(x,y)Ψ=aΨm1​,n1​​(x,y)+bΨm2​,n2​​(x,y)+cΨm3​,n3​​(x,y). We first find the trajectories of the nodal points for different combinations of the quantum numbers m,nm,nm,n. Then we study, in detail, a case with relatively large quantum numbers and two equal m′sm'sm′s. We find %We find first the nodal points where Ψ=0Ψ=0Ψ=0. The nodes can be found analytically only if mmm and nnn are small. If two m′sm'sm′s (or two n′sn'sn′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.
We analyze the dynamics of a quantum particle in a one-dimensional bistable potential within the framework of Bohm's quantum mechanics. We give arguments that evidence the fallacy of certain claims found in the literature dealing with the impossibility of chaotic behavior of Bohmian trajectories in one-dimensional systems. We find that an appropriate choice for the initial position and wave packet causes the particle to undergo periodic, quasiperiodic, or chaotic motion. The transitions between these regimes occur in a continuos fashion.
Bohmian mechanics is a causal interpretation of quantum mechanics in which particles describe trajectories guided by the wave function. The dynamics in the vicinity of nodes of the wave function, usually called vortices, is regular if they are at rest. However, vortices generically move during time evolution of the system. We show that this movement is the origin of chaotic behavior of quantum trajectories. As an example, our general result is illustrated numerically in the two-dimensional isotropic harmonic oscillator. © EDP Sciences.
Bohmian mechanics is a causal interpretation of quantum mechanics in which there are well-defined particle trajectories guided by the wave function. Within this framework, it is shown that certain classical integrable systems can exhibit both periodic and chaotic behaviours. In this paper, we extend the work of Borondo et al. by investigating the behaviours of commensurate two-dimensional harmonic oscillator systems with different ratios of frequencies. We show that the system may produce chaotic behavior of Bohmian trajectories to be dependenton frequencies ratio. This result is illustrated numerically in computer experiments displaying the Bohmian trajectories.