GENERALIZED TIME-ENERGY UNCERTAINTY RELATIONS AND BOUNDS ON LIFETIMESRESONANCES

A precise form of the quantum-mechanical time-energy uncertainty relat ion is derived. For any given initial state (density operator), time-d ependent Hamiltonian, and subspace of reference states, it gives upper and lower bounds for the probability of finding the system in a state in that. subspace at a later or earlier time. The bounds involve only the initial data, the energy uncertainty in the initial state, and th e energy uncertainty in the reference subspace. They describe how fast the state enters or leaves the reference subspace. They are exact if, but not only if, the initial state or the projection onto the referen ce subspace commutes with the Hamiltonian. The basic tool used in the proof is a simple inequality for expectation values of commutators, wh ich generalizes the usual uncertainty relation. By introducing suitabl e comparison dynamics (trial propagators), the bounds can be made arbi trarily tight. They represent a time-dependent variational principle, in terms of trial propagators, which provides explicit error estimates and reproduces the exact time evolution when one varies over ail tria l propagators. As illustrations, we derive accurate lower bounds on th e escape time of a particle out of a potential well modeling a quantum dot, and the total time before which a He+ ion moving in a uniform ma gnetic field loses its electron.