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Journal of Physics A Mathematical and General

The basis generator method: optimized dynamical representation of the solution of time-dependent quantum problems

O.J. Kroneisen, Hans Jürgen Lüdde, Tom Kirchner, R. M. Dreizler

Published Jan 1, 1999• 76 citations
Publisher

Abstract

The theoretical investigation of time-dependent quantum systems requires the solution of the time-dependent Schrödinger (Dirac) equation. The basis generator method presented here allows a systematic construction of dynamically adapted wavefunctions based on a decomposition of the Hilbert space into a hierarchical structure of finite subspaces. For the class of interactions obeying an inverse integer power law, e.g., Coulomb and polarization interactions, an explicit representation of the dynamically optimized basis set is given.

DOI
10.1088/0305-4470/32/11/009

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The theoretical investigation of time-dependent quantum systems requires the solution of the time-dependent Schrödinger (Dirac) equation. The basis generator method presented here allows a systematic construction of dynamically adapted wavefunctions based on a decomposition of the Hilbert space into a hierarchical structure of finite subspaces. For the class of interactions obeying an inverse integer power law, e.g., Coulomb and polarization interactions, an explicit representation of the dynamically optimized basis set is given.
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Krylov subspace methods in quantum dynamics identify the minimal subspace in which a process unfolds. To date, their use is restricted to time evolutions governed by time-independent generators. We introduce a generalization valid for driven quantum systems governed by a time-dependent Hamiltonian that maps the evolution to a diffusion problem in a one-dimensional lattice with nearest-neighbor hopping probabilities that are inhomogeneous and time dependent. This representation is used to establish a novel class of fundamental limits to the quantum speed of evolution and operator growth. We also discuss generalizations of the algorithm, adapted to discretized time evolutions and periodic Hamiltonians, with applications to many-body systems.
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All bound-state solutions of the Dirac Coulomb problem are shown to provide a Hilbert space for irreducible representations of the de Sitter group O(4,1)O(4, 1)O(4,1) Selected solutions of this problem provide the basis for other representations of O(4,1)O(4, 1)O(4,1) These results are compared to analogous results previously obtained for the Schr\"odinger Coulomb problem. Operator forms of the generators of O(4,1)O(4, 1)O(4,1) are explicitly constructed as functions of p and r which operate on the Hilbert space provided by the Dirac problem.