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Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

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Time in Quantum Mechanics - Vol. 2

Part of the book series: Lecture Notes in Physics ((LNP,volume 789))

Abstract

Interest in optimal time evolution dates back to the end of the seventeenth century, when the famous brachistochrone problem was solved almost simultaneously by Newton, Leibniz, l‘Hôpital, and Jacob and Johann Bernoulli. The word brachistochrone is derived from Greek and means shortest time (of flight). The classical brachistochrone problem is stated as follows: A bead slides down a frictionless wire from point A to point B in a homogeneous gravitational field. What is the shape of the wire that minimizes the time of flight of the bead? The solution to this problem is that the optimal (fastest) time evolution is achieved when the wire takes the shape of a cycloid, which is the curve that is traced out by a point on a wheel that is rollingon flat ground.

The shortest path between two truths in the real domain passes through the complex domain.

Jacques Hadamard, The Mathematical Intelligencer 13 (1991)

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Acknowledgement

We have benefited greatly from many discussions with Drs. U. Güunther and B. Samsonov. We thank Dr. D.W. Hook for his assistance in preparing the figures used in this chapter. CMB is supported by a grant from the US Department of Energy.

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Authors and Affiliations

  1. Department of Physics, Washington University, 63130, St. Louis, MO, USA

    Carl M. Bender

  2. Department of Mathematics, Imperial College London, SW7 2AZ, London, UK

    Dorje C. Brody

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  1. Carl M. Bender
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Correspondence to Carl M. Bender .

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Editors and Affiliations

  1. Depto. Quimica Fisica EHU, Universidad Pais Vasco, Apartado 644, Bilbao, 48080, Spain

    Gonzalo Muga

  2. Inst. Mathematische Physik, TU Braunschweig, Mendelssohnstr. 3, Braunschweig, 38106, Germany

    Andreas Ruschhaupt

  3. Inst. Mathematical Sciences, Imperial College London, Prince's Gate 53, London, SW7 2PG, UK

    Adolfo del Campo

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Bender, C.M., Brody, D.C. (2009). Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians. In: Muga, G., Ruschhaupt, A., del Campo, A. (eds) Time in Quantum Mechanics - Vol. 2. Lecture Notes in Physics, vol 789. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03174-8_12

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  • DOI: https://doi.org/10.1007/978-3-642-03174-8_12

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