Upper bounds for the energy expectation in time-dependent quantum mechanics

Peter Pfeifer, Jürg Martin Fröhlich

A precise form of the quantum-mechanical time-energy uncertainty relation is derived. For any given initial state (density operator), time-dependent 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 the 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 reference subspace commutes with the Hamiltonian. The basic tool used in the proof is a simple inequality for expectation values of commutators, which generalizes the usual uncertainty relation. By introducing suitable comparison dynamics (trial propagators), the bounds can be made arbitrarily 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 all trial propagators. As illustrations, we derive accurate lower bounds on the escape time of a particle out of a potential well modeling a quantum dot, and the total time before which a ${\mathrm{He}}^{+}$ ion moving in a uniform magnetic field loses its electron.

Jean-Marie Barbaroux, Alain Joye

Takeshi Morita

Abstract The range of motion of a particle with certain energy E confined in a potential is determined from the energy conservation law in classical mechanics. The counterpart of this question in quantum mechanics can be regarded as what is the possible range of expectation values of the position operator 〈x〉 of a particle that satisfies E = 〈H〉. This range depends on the state of the particle, but the universal upper and lower bounds, which are independent of the state, must exist. In this study, we show that these bounds can be derived by using the bootstrap method. We also point out that the bootstrap method can be regarded as a generalization of the uncertainty relations, meaning that the bounds are determined by the uncertainty relations in a broad sense. Furthermore, the bounds on possible expectation values of various quantities other than position can be determined in the same way. However, in the case of multiple identical particles (bosons and fermions), we find some difficulty in the bootstrap method. Because of this issue, the predictive power of the bootstrap method in multi-particle systems is limited in the derivation of observables including energy eigenstates. In addition, we argue an application of the bootstrap method to thermal equilibrium states. We find serious issues that temperature and entropy cannot be handled. Although we have these issues, we can derive some quantities in micro-canonical ensembles of integrable systems governed by generalized Gibbs ensembles.

Pulak Ranjan Giri

M. Cohen, Tova Feldmann

A simple unified procedure, based on a generalization of a method first used by Weinstein in 1934, yields effective lower and upper bounds to energy levels and overlap integrals of quantum-mechanical systems. A further generalization yields bounds to expectation values.