Year: 1992
Type: article
Abstract: 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
Source: Computers in Physics
DOI: 10.1063/1.4823048
Authors: Niraj Srivastava, Charles Kaufman, Gerhard Müller
Institution: University of Rhode Island
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4250311590