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arXiv:2411.12472 · Particles

Classical and Bohmian Trajectories in Integrable and Nonintegrable Systems

G. Contopoulos, Athanasios C. Tzemos

Published Nov 25, 2024• 2 citations
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Abstract

In the present paper, we study both classical and quantum Hénon–Heiles systems. In particular, we make a comparison between the classical and quantum trajectories of integrable and nonintegrable Hénon–Heiles Hamiltonians. From a classical standpoint, we study both theoretically and numerically the form of invariant curves in the Poincaré surfaces of section for several values of the coupling parameter in the integrable case and compare them with those in the nonintegrable case. Then, we examine the corresponding Bohmian trajectories, and we find that they are chaotic in both cases, but with chaos emerging at different times.

DOI
10.3390/particles7040064

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In the present paper, we study both classical and quantum Hénon–Heiles systems. In particular, we make a comparison between the classical and quantum trajectories of integrable and nonintegrable Hénon–Heiles Hamiltonians. From a classical standpoint, we study both theoretically and numerically the form of invariant curves in the Poincaré surfaces of section for several values of the coupling parameter in the integrable case and compare them with those in the nonintegrable case. Then, we examine the corresponding Bohmian trajectories, and we find that they are chaotic in both cases, but with chaos emerging at different times.
The evolution of wave packets under the influence of a Hénon–Heiles potential has been investigated by direct numerical solution of the time-dependent Schrödinger equation. Coherent state Gaussians with a variety of mean positions and momenta were selected as initial wave functions. Three types of diagnostics were used to identify chaotic behavior, namely, phase space trajectories computed from the expected values of coordinates and momenta, the correlation function P(t)=‖〈ψ(0)‖ψ(t)〉‖2, and the uncertainty product or phase space volume V(t)=ΔxΔyΔpxΔpy. The three approaches lead to a consistent interpretation of the system’s behavior, which tends to become more chaotic as the energy expectation value of the wave packet increases. The behavior of the corresponding classical system, however, is not a reliable guide to regular or chaotic behavior in the quantum mechanical system.
This paper presents a number of numerical investigations of orbits in the de Broglie–Bohm version of quantum mechanics. We first clarify how the notion of chaos should be implemented in the case of Bohmian orbits. Then, we investigate the Bohmian orbits in three different characteristic quantum systems: (a) superposition of three stationary states in the Hamiltonian of two uncoupled harmonic oscillators with incommensurable frequencies, (b) wave packets in a Hénon–Heiles-type Hamiltonian and (c) a modified two-slit experiment. In these examples, we identify regular or chaotic orbits and also orbits exhibiting a temporarily regular and then chaotic behaviour. Then, we focus on a numerical investigation of the Bohm–Vigier (Bohm and Vigier 1954 Phys. Rev. 26 208) theory, that an arbitrary initial particle distribution P should asymptotically tend to |ψ| 2 , by considering the role of chaotic mixing in causing irregularity of Madelung's flow, a necessary condition for P to tend to |ψ| 2 . We find that the degree of chaos of a particular system correlates with the speed of convergence of P to |ψ| 2 . In the case of wave-packet dynamics, our numerical data show that the time of convergence scales exponentially with the inverse of the effective perturbation from the harmonic oscillator Hamiltonian. The latter result can be viewed as a quantum analogue of Nekhoroshev's (Nekhoroshev 1977 Russ. Math. Surveys 32 1) theorem of exponential stability in classical nonlinear Hamiltonian dynamics.
Quantum domain behaviour of classically chaotic systems is studied using the quantum theory of motion in teh sense of clasical interpretationof quantum mechanics as developed by de Broglie and Bohm. Dynamics of quantum Henon- Heiles oscillatorm barbanis oxcillatior and CTW oscillator are ananysed with the help of quantum Lypunov exponent and Kolmogorov-Sinai entropy defined in terms of the distance between two initially close Bohmian trajectories. Standard diagnostics of quantum chaos like auticorrelation function and the assocated power spectrum, nearest neighbour spacing distribution phase space volume, spectral rigidity, etc. support these results. Quantum theory of motion provides an alternative route for understanding quantum chaos. Nonlinear dynamics of integrable systems in quantum domain is also properly taken care of within this framework.
Abstract Chaos in Bohmian quantum mechanics is an open field of research. In general, most of the 3D Bohmian trajectories are free to wander around the 3D space. However, there are cases where the evolution of the trajectories is dictated by exact or approximate integrals of motion. A first case corresponds to partial integrability, where the trajectories (ordered and chaotic) evolve on certain integral surfaces. A second case corresponds to ordered trajectories. In this paper we extend our previous work in 3D Bohmian chaos by using both forms of integrability and discuss their physical implications.
The quantum-classical correspondence is studied for a periodically driven quartic oscillator exhibiting integrable and chaotic dynamics, by studying the Bohmian trajectory of the corresponding "classical" Schrödinger equation. Phase plots and the Kolmogorov-Sinai entropy are computed and compared with the classical trajectory as well as the Bohmian trajectory obtained from the time dependent Schrödinger equation. Bohmian mechanics at the classical limit appears to mimick the behavior of a dissipative dynamical system.
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.