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Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

  • Carl M. BenderEmail author
  • Dorje C. Brody
Chapter
Part of the Lecture Notes in Physics book series (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.

Keywords

Bloch Sphere Unitary Motion Quantum Mechanical Theory Ghost State Unitary Time Evolution 
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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Notes

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of PhysicsWashington UniversitySt. LouisUSA
  2. 2.Department of MathematicsImperial College LondonLondonUK

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