Lower bound of minimal time evolution in quantum mechanics

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\times 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 \bf P of the evolving quantum state ρ={1/2}(\bf 1+ \bf{P}\cdot\boldsymbolσ) with the vector \boldsymbol{\mathcal O}(Θ) of the 2\times 2 hermitian Hamiltonians H ={1/2}({\mathcal O}_0\boldsymbol{1}+ \boldsymbol{\mathcal O}(Θ)\cdot\boldsymbolσ) and it is shown that the Hamiltonian H can be parameterized by two independent parameters {\mathcal O}_0 and Θ.

4 pages, no figure, revtex4

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Cited by in corpus (5)

• Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control
• Bypassing the bandwidth theorem with PT symmetry
• Time-dependent Hamiltonians with 100% evolution speed efficiency
• The PT-symmetric brachistochrone problem, Lorentz boosts and non-unitary operator equivalence classes
• Some Remarks on Quantum Brachistochrone