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Related Experiment Videos

Phase space deformation and basis set optimization.

Matthew C Cargo1, Robert G Littlejohn

  • 1Department of Physics, University of California, Berkeley, California 94720, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 28, 2002
PubMed
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Basis set size reduction is achieved by deforming phase space regions. One-dimensional problems show significant reduction (B/C=1+O(Planck

Area of Science:

  • Quantum mechanics
  • Computational physics
  • Mathematical physics

Background:

  • Finding unknown eigenfunctions often requires large basis sets, increasing computational cost.
  • Deforming phase space regions is a theoretical approach to simplify complex quantum systems.
  • Basis set size (B) and the number of converged eigenfunctions (C) are key metrics in computational quantum mechanics.

Purpose of the Study:

  • To investigate the effectiveness of phase space deformation for reducing basis set size in quantum mechanics.
  • To analyze the achievable reduction in one-dimensional and higher-dimensional problems.
  • To provide theoretical and numerical evidence for the proposed method.

Main Methods:

  • Deformation of a given phase space region into a standard, integrable region.

Related Experiment Videos

  • Theoretical analysis of basis set size reduction in one and higher dimensions.
  • Numerical confirmation of theoretical predictions, including convergence rates.

    • Main Results:

    • In one-dimensional problems, basis set size reduction achieves B/C = 1 + O(Planck's constant).
    • Numerical examples demonstrate exponential convergence with increased basis set size in 1D.
    • In higher dimensions, a significant reduction is proven impossible, with B/C = a + o(1) where a > 1.

    Conclusions:

    • Phase space deformation offers effective basis set reduction, particularly in one-dimensional quantum systems.
    • The achievable reduction in higher dimensions is limited, with a geometric lower bound.
    • The method provides a pathway to more efficient computation of eigenfunctions in specific quantum mechanical problems.