Lower bound of minimal time evolution in quantum mechanics
arXiv:0706.3653 · doi:10.1007/s10773-008-9650-0
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