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Simple Quantum Systems

  • Philipp O. J. Scherer
Part of the Graduate Texts in Physics book series (GTP)

Abstract

We discuss several approaches to solve the one-dimensional Schrödinger equation numerically. The dispersion of simple finite differences deviates largely from the exact relation unless high order differences are used. More accurate pseudo-spectral methods evaluate the kinetic energy part in Fourier space. The time evolution can be approximated by unitary rational expressions like Cauchy’s form. Multistep differencing schemes have comparable accuracy but are explicit. The split operator approximation of the time evolution operator leads to the real-space product formula. In a computer experiment we simulate a one-dimensional wave packet.

Subsequently we study a two-state system in an oscillating field, a three-state system as a model for superexchange, the Landau-Zener model for curve-crossing and the ladder model for exponential decay. The density matrix formalism is used to describe a dissipative two-state system in analogy to Bloch’s equations for nuclear magnetic resonance. Computer experiments simulate resonance behavior, saturation and power broadening. The generation of a coherent superposition state or a spin flip are simulated and discussed in connection with the manipulation of a qubit.

Keywords

Density Matrix Wave Packet Time Evolution Operator Bloch Equation Gaussian Wave Packet 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Philipp O. J. Scherer
    • 1
  1. 1.Physikdepartment T38Technische Universität MünchenGarchingGermany

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