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Related Experiment Videos

Semiclassical limit for a truncated Hamiltonian.

Cesar Castilho1, A. M. Ozorio De Almeida

  • 1Mathematics Department, University of California at Santa Cruz, Santa Cruz, California 95064Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud, 150, Rio de Janeiro, RJ 22290-180, Brazil.

Chaos (Woodbury, N.Y.)
|June 1, 1996
PubMed
Summary

Numerical calculations of polynomial Hamiltonians reveal basis set cutoff dependence. Semiclassical periodic orbit theory explains spectral features by relating finite matrices to classical "Action Billiards".

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Area of Science:

  • Quantum mechanics
  • Mathematical physics
  • Chaos theory

Background:

  • Numerical calculations of polynomial Hamiltonians often show basis set cutoff dependence.
  • Understanding the full energy spectrum requires advanced analytical techniques.

Purpose of the Study:

  • To explain spectral features of polynomial Hamiltonians using semiclassical methods.
  • To analyze the relationship between finite Hamiltonian matrices and classical dynamical systems.

Main Methods:

  • Numerical calculation of eigenenergies for polynomial Hamiltonians.
  • Analysis of finite Hamiltonian matrices as classical "Action Billiards".
  • Application of semiclassical periodic orbit theory.

Main Results:

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  • Identified dependence of eigenenergies on basis set cutoff.
  • Explained spectral features using semiclassical periodic orbit theory and the "Action Billiard" model.
  • Observed interference of low-period orbits at higher energies.
  • Found the billiard system becomes more regular than the untruncated Hamiltonian at higher energies, indicated by the Berry-Robnik level spacing distribution.

Conclusions:

  • Semiclassical periodic orbit theory provides insights into the spectral properties of polynomial Hamiltonians.
  • The "Action Billiard" model effectively explains spectral features related to basis set cutoffs.
  • System regularity can change with energy, deviating from the untruncated Hamiltonian behavior.