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Lower Bound of Minimal Time Evolution in Quantum Mechanics

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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.

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  1. Theory Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta, 700064, India

    Pulak Ranjan Giri

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  1. Pulak Ranjan Giri
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Correspondence to Pulak Ranjan Giri.

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Giri, P.R. Lower Bound of Minimal Time Evolution in Quantum Mechanics. Int J Theor Phys 47, 2095–2100 (2008). https://doi.org/10.1007/s10773-008-9650-0

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  • DOI: https://doi.org/10.1007/s10773-008-9650-0

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