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arXiv:quant-ph/0212093

Quantum Chaos via the Quantum Action

H. Kröger

Published Dec 16, 2002• 1 citation
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Abstract

We discuss the concept of the quantum action with the purpose to characterize and quantitatively compute quantum chaos. As an example we consider in quantum mechanics a 2-D Hamiltonian system - harmonic oscillators with anharmonic coupling - which is classically a chaotic system. We compare Poincaré sections obtained from the quantum action with those from the classical action.

DOI
10.48550/arxiv.quant-ph/0212093

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We discuss the concept of the quantum action with the purpose to characterize and quantitatively compute quantum chaos. As an example we consider in quantum mechanics a 2-D Hamiltonian system - harmonic oscillators with anharmonic coupling - which is classically a chaotic system. We compare Poincaré sections obtained from the quantum action with those from the classical action.
We have recently suggested a quantum action, which has the form of a classical action and takes into account quantum effects via renormalized action parameters. Here we apply it to quantum chaos. We study a system in 2-D with weak anharmonic coupling (Vcoupl∝x2y2V_{coupl} \propto x^{2}y^{2}Vcoupl​∝x2y2) being classically chaotic. We construct the quantum action at finite temperature. We compute Poincaré sections of the quantum action at that temperature and compare it with those of the classical action. We observe chaotic behavior in both cases.
Does quantum chaos exist?If it exists, what is it?If it does not, what was it supposed to have been?Why should we associate chaos with a "third revolution in physics" if it fails to be expressible in terms of quantum mechanics, our most fundamental theory of physical reality?Is there something wrong with quantum mechanics?Or is chaos merely a mathematical construct relevant only for models in classical mechanics?Is the human mind doomed to interpret and understand quantum mechanics in classical terms?The list of unanswered questions in quantum chaos research itself shows symptoms of the phenomenon it attempts to grasp. 1 How do we recognize classical Hamiltonian chaos?We have dealt with this question in our two previous columns.2,3 In Part I we discussed the implications of integrability and non-integrability for the phase-space trajectories of classical Hamiltonian systems with two degrees of freedom.We described the method of Poincaré surfaces of section as a convenient and striking discriminant between the two possibilities.In Part II we searched for an alternate mode of representation for classical Hamiltonian systems, a mode that is an equally powerful indicator of chaos, but one that can be more directly translated into quantum mechanics than the Poincaré map.We proposed a representation based on the construction of classical invariants via time averages of dynamical variables, a representation that can be employed under very general circumstances including integrable and nonintegrable models.In this column we shall construct quantum invariants, demonstrate the impact of non-integrability on these quantities, and discuss their properties in relation to their classical counterparts.In a future column we intend to introduce quantum Poincaré surfaces of section for stationary states and bring full circle our survey of quantum and classical Hamiltonian chaos.Consider a quantized integrable model.Its eigenstates are labeled by quantum numbers.There are as many quantum numbers as there are degrees of freedom.Each quantum number represents a quantized action.In the still evolving language of quantum chaos, quantum numbers are said to have been "lost" in the transition from a quantum system whose classical counterpart is integrable to one whose counterpart is chaotic.The broken, or lost, tori of classical chaos become lost quantum numbers of quantum chaos.As we shall see, the loss of quantum numbers is vividly mirrored in the behavior of the quantum invariants.We begin by describing the construction of the quantum invariants for a
The chaos exhibited by a quantum mechanical system is revealed in various aspects such as level spacing, avoiding of level crossing on changing a parameter value, irregular nodal lines of wave function, random matrix and so on. On the other hand since the wave function, the propagator or any physical quantity represented by the average of a phase function is usually quasi-periodic in time development, no exponential divergence of its value takes place between the neighboring two initial values. Furthermore the energy of a kicked oscillator ceases to increase after a certain period of time. These observations lead one to suspect the existence of chaos in quantal systems. It is shown that the nature of quasi-periodicity in quantum system is shared with the distribution function of classical hamiltonian system. The action integral which appears in the formulation of the propagator is the key quantity which can connect the classical and quantal systems, and is considered as serving to search for the origin of quantum chaos. The roles of the action integral in the determination of the energy levels and the behavior of the wave function are discussed.
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.