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Simple Quantum Systems

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Computational Physics

Part of the book series: Graduate Texts in Physics ((GTP))

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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.

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Notes

  1. 1.

    For instance collisions or the electromagnetic radiation field.

  2. 2.

    If, for instance the wavefunction depends on the coordinates of N particles, the scalar product is defined by \(\langle \psi_{n}|\psi_{n'}\rangle = \int d^{3}r_{1}\cdots d^{3}r_{N}\psi_{n}^{*}(r_{1}\cdots r_{N})\psi _{n'}(r_{1}\cdots r_{N})\).

  3. 3.

    They are often called the “coherence” of the two states.

  4. 4.

    The Padé approximation (Sect. 2.4.1) of order [1,1].

  5. 5.

    For simplicity only the case of even M is considered.

  6. 6.

    This basis is usually incomplete.

  7. 7.

    We assume E 2>E 1.

  8. 8.

    So called Rabi oscillations.

  9. 9.

    The matrix of this system corresponds to the Liouville operator.

  10. 10.

    We assume Δ≥0, such that the equilibrium value of z=ρ 11−ρ 22 is negative. Eventually, the two states have to be exchanged.

  11. 11.

    For instance T 2=2T 1 for pure radiative damping.

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Authors and Affiliations

  1. Physikdepartment T38, Technische Universität München, Garching, Germany

    Philipp O. J. Scherer

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Scherer, P.O.J. (2013). Simple Quantum Systems. In: Computational Physics. Graduate Texts in Physics. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00401-3_21

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