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Emergence of Quantum Chaos in a Model 4-Body System

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Abstract

We implement the method of ``slow variable discretization'' (SVD) to solve the Schrodinger equation for the (quasi)-bound state energy levels of a system of four interacting atoms moving in one spatial dimension. The SVD method we implement treats the hyper-radius (a measure of the overall size of the system) as a ``slowly varying'' parameter. This parameter can then be discretized using the ``discrete variable representation'' (DVR) technique, which guarantees our numerical approximations will converge to being exact given a large enough basis size. To measure chaos, we appeal to the Bohigas, Giannoni, and Schmidt conjecture, which states a classically chaotic system can be statistically described by random matrix theory in quantum mechanics, where random matrix theory makes predictions for the nearest neighbor energy level spacings. We calculate the distribution of energy-level spacings and fit it to the Brody distribution to extract a Brody parameter. A calculation in the adiabatic hyperspherical representation in which all nonadiabatic couplings are ignored gives a perfectly Poisson distribution, corresponding to a Brody parameter of zero. A calculation with SVD, on the other hand, which implicitly incorporates nonadiabatic couplings, yields a Brody parameter of 0.571, indicating a sizable degree of level repulsion characteristic of the transition from an uncorrelated system to one with chaos.

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Authors

  • Cooper Johnson

    Trinity University

  • Eric Davis

    University of Texas at Arlington, Sultan Qaboos University, UTA HEP group, Department of Integrated Bio \& Nano Systems, University of Houston, Houston, TX 77204, USA, Department of Electrical and Computer Engineering, Nano Tech Center, Texas Tech University, University of Michigan, University of Waerloo, Canada, Texas Tech University, Dept of Mechanical Eng, TTU; Dept. of Physics and Astronomy, TTU, Dept of Cell Biology and Biochemistry, TTU Health Sci Ctr, Dept of Physics and Astronomy, TTU, Dept of Mechanical Engineering, TTU, Dept of Physics and Astronomy, TTU; Sch of Health and Sci, Purdue Univ, Dept of Mathematical and Systems Engineering, Shizuoka Univ, Department of Physics, University of Texas at Dallas, Richardson, TX 75080, Department of Physics, University of Texas at Dallas, Richardson, TX 75080., LeTourneau University, None, University of Waterloo, Canada, Texas Tech Univ, Trinity University, the University of Texas at Dallas, Texas Tech University, Lubbock, TX, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada, University of Dallas, Liverpool John Moores University, University Of Houston, Biolog Department, TCU, Home Schooled high school student, Los Alamos National Laboratory