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International Journal of Theoretical Physics

, Volume 47, Issue 7, pp 2095–2100 | Cite as

Lower Bound of Minimal Time Evolution in Quantum Mechanics

  • Pulak Ranjan Giri
Article

Abstract

We show that the total time of evolution from the initial quantum state to final quantum state and then back to the initial state, i.e., making a round trip along the great circle over S 2, must have a lower bound in quantum mechanics, if the difference between two eigenstates of the 2×2 Hamiltonian is kept fixed. Even the non-hermitian quantum mechanics can not reduce it to arbitrarily small value. In fact, we show that whether one uses a hermitian Hamiltonian or a non-hermitian, the required minimal total time of evolution is same. It is argued that in hermitian quantum mechanics the condition for minimal time evolution can be understood as a constraint coming from the orthogonality of the polarization vector P of the evolving quantum state \(\rho=\frac{1}{2}(\boldsymbol{1}+\mathbf{P}\cdot\boldsymbol{\sigma})\) with the vector \(\boldsymbol{\mathcal{O}}(\Theta)\) of the 2×2 hermitian Hamiltonians \(H=\frac{1}{2}({\mathcal{O}}_{0}\boldsymbol{1}+\boldsymbol{\mathcal{O}}(\Theta)\cdot\boldsymbol{\sigma})\) and it is shown that the Hamiltonian H can be parameterized by two independent parameters \({\mathcal{O}}_{0}\) and Θ.

Keywords

Quantum brachistochrone Lower bound on time evolution Hermitian quantum mechanics Nonhermitian quantum mechanics 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Theory DivisionSaha Institute of Nuclear PhysicsCalcuttaIndia

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