nt - p h / 03 03 08 5 v 1 13 M ar 2 00 3 The quantum speed limit

The Hamiltonian H is the generator of the dynamics of physical systems. In this respect, the energy content of a system will determine its evolution time scales. For instance, the time-energy uncertainty relation states that the time it takes for a system to evolve to an orthogonal configuration in the Hilbert space is limited by the inverse of the energy spread of the system. Analogously, the Margolus-Levitin theorem relates the same quantity to the mean energy. More generally, one can consider the minimum time required for a system to “rotate” from its initial configuration by a predetermined amount. In Ref. 5 we have shown that also this time is limited by the energy and energy spread. More specifically, given a value of ǫ ∈ [0, 1], consider the time t it takes for a system in the state |Ψ〉 to evolve to a state |Ψ(t)〉 such that P (t) ≡ |〈Ψ|Ψ(t)〉| = ǫ . (1)