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Local quantum ergodic conjecture.

Eduardo Zambrano1, W P Karel Zapfe2, Alfredo M Ozorio de Almeida2

  • 1Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany.

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Summary
This summary is machine-generated.

This study redefines the quantum ergodic conjecture using the chord function, revealing universal patterns in chaotic quantum systems. Numerical evidence supports these findings, highlighting exceptions in scarred eigenstates.

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Area of Science:

  • Quantum mechanics
  • Statistical mechanics
  • Chaos theory

Background:

  • The quantum ergodic conjecture relates Wigner functions of chaotic Hamiltonians to delta functions on the energy shell.
  • This connection aids classical ergodic expectation calculations but faces theoretical challenges with the Wigner function's properties.

Purpose of the Study:

  • To address limitations of the standard quantum ergodic conjecture by reformulating it in terms of the chord function.
  • To investigate the local properties of quantum eigenstates in classically chaotic systems.

Main Methods:

  • Fourier transform of the Wigner function to obtain the chord function.
  • Analysis of information within a Planck volume in the phase space of chords.
  • Numerical simulations using a Hamiltonian exhibiting soft chaos.

Main Results:

  • The reformulated conjecture, focusing on the chord function, requires only local information in phase space.
  • A universal pattern of orthogonality loci is predicted for nearby translations of ergodic eigenstates within a narrow energy range.
  • Numerical evidence supports the universality, with scarred eigenstates identified as exceptions.

Conclusions:

  • The chord function provides a more robust framework for the quantum ergodic conjecture.
  • The study reveals universal properties of quantum chaotic systems, with implications for understanding quantum-classical correspondence.
  • Scarred eigenstates represent a distinct class of quantum states that deviate from the universal ergodic pattern.