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Symmetry

The Emergence of Chaos in Quantum Mechanics

Emilio Fiordilino

Published May 8, 2020• 5 citations
Publisher

Abstract

Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difficult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser field assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schrödinger equation is solved within the two-state approximation; the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular filling of the Poincaré map and exponential separation of the trajectories of the Bloch vector σ in the Bloch sphere.

DOI
10.3390/sym12050785

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Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difficult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser field assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schrödinger equation is solved within the two-state approximation; the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular filling of the Poincaré map and exponential separation of the trajectories of the Bloch vector σ in the Bloch sphere.
A new model is proposed for the purpose of modelling the ``wave function collapse'' of a two-state quantum system. The collapse to a classical state is driven by a nonlinear evolution equation with an extreme sensitivity to absolute phase. It is hypothesized that the phase, or part of it, is displaying chaotic behaviour. This chaotic behaviour can then be responsible for the apparent indeterminacy we are experiencing for a single quantum system. Through this randomness, the statistical ``ensemble'' behaviour, due to Born, to describe a single quantum system, is no longer needed.
Dynamical chaos has recently been shown to exist in the Gaussian approximation in quantum mechanics and in the self-consistent mean field approach to studying the dynamics of quantum fields. In this study, we first note that any variational approximation to the dynamics of a quantum system based on the Dirac action principle leads to a classical Hamiltonian dynamics for the variational parameters. Since this Hamiltonian is generically nonlinear and nonintegrable, the dynamics thus generated can be chaotic, in distinction to the exact quantum evolution. We then restrict our attention to a system of two biquadratically coupled quantum oscillators and study two variational schemes, the leading order large-NNN (four canonical variables) and Hartree (six canonical variables) approximations. The chaos seen in the approximate dynamics is an artifact of the approximations: this is demonstrated by the fact that its onset occurs on the same characteristic time scale as the breakdown of the approximations when compared to numerical solutions of the time-dependent Schr\"odinger equation.
In contrast to the classical Hamiltonian mechanics whose chaotic behaviors are now well established, quantum mechanics is still obscure about the origin of its random behavior. It is shown in this paper that in case of the stadium billiard, the fact that the nodal lines are smooth but are randomly oscillating, as first described by Heller, and MacDonald and Kaufman, can be interpreted as due to the irregular pseudochaotic behavior of the phase. To do this, we employ the path integral representation of propagator and the reinterpreted Bohm's formulation of quantum mechanics. Thus we reach the conclusion that although the chaotic behavior in quantum mechanics is suppressed by the Einstein quantization condition, it is observed in the properly defined phase of the propagator, or of the wave function, as a pseudochaos. In other words quantum mechanics has the counterpart of classical chaos in the phase of the propagator, and consequently, the wave function is also, pseudochaotic (random) with respect to q , but it does not exhibit exponential instability. The essential feature of quantum chaos manifests itself in the vanishings of the average of the wave function and the two-point correlation of the wave function at high energies. They are closely related to ergodicity and quantum measurement, and in particular new light will be shed on the measurement theory. The basic assumptions in the random matrix theory are discussed, confirming the universality of level spacing distribution.
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.
We describe a quantum many-body system undergoing multiple resonant tunneling which exhibits chaotic behavior in numerical simulations of a mean field approximation. This phenomenon, which has no counterpart in the classical limit, is due to effective nonlinearities in the tunneling process and can be observed in principle within a heterostructure. 03.65.-w, 05.45.+b, 73.40.Gk Typeset using REVT E X In current studies the problem of quantum chaos is usually posed in the following terms. We are given a system which exhibits chaotic behavior classically: does this property survive quantization? Typically one studies the spectrum of the quantum mechanical Hamiltonian operator. However spectral differences at the quantum level between classically integrable and chaotic systems do not seem to imply dramatic differences in the time evolution of the quantum mechanical observables. Relics of classical chaos exist in low dimensional systems in short wavelength or large quantum numbers limits ...
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.