Problem Of Time In Quantum Mechanics Research Articles

A review of the decoherent histories approach to the arrival time problem in quantum theory

We review recent progress in understanding the arrival time problem in quantum mechanics, from the point of view of the decoherent histories approach to quantum theory. We begin by discussing the arrival time problem, focussing in particular on the role of the probability current in the expected classical solution. After a brief introduction to decoherent histories we review the use of complex potentials in the construction of appropriate class operators. We then discuss the arrival time problem for a particle coupled to an environment, and review how the arrival time probability can be expressed in terms of a POVM in this case. We turn finally to the question of decoherence of the corresponding histories, and we show that this can be achieved for simple states in the case of a free particle, and for general states for a particle coupled to an environment.

Quantum arrival time for open systems

We extend previous work on the arrival time problem in quantum mechanics, in the framework of decoherent histories, to the case of a particle coupled to an environment. The usual arrival time probabilities are related to the probability current, so we explore the properties of the current for general open systems that can be written in terms of a master equation of Lindblad form. We specialise to the case of quantum Brownian motion, and show that after a time of order the localisation time the current becomes positive. We show that the arrival time probabilities can then be written in terms of a POVM, which we compute. We perform a decoherent histories analysis including the effects of the environment and show that time of arrival probabilities are decoherent for a generic state after a time much greater than the localisation time, but that there is a fundamental limitation on the accuracy, $\delta t$, with which they can be specified which obeys $E\delta t>>\hbar$. We confirm that the arrival time probabilities computed in this way agree with those computed via the current, provided there is decoherence. We thus find that the decoherent histories formulation of quantum mechanics provides a consistent explanation for the emergence of the probability current as the classical arrival time distribution, and a systematic rule for deciding when probabilities may be assigned.

Time in classical and in quantum mechanics

This paper presents an analysis of the time concept in classical mechanics from the perspective of the invariants of a motion. The analysis shows that there is a conceptual gap concerning time in the Dirac–Heisenberg–von Neumann formalism and that Bohr's complementarity principle does not fill the gap. In the Dirac–Heisenberg–von Neumann formalism, a particle's properties are represented by Heisenberg matrices. This axiom is the source of the time problem in quantum mechanics.

Time operator in quantum mechanics

There have been great theoretical and practical interests in investigating time problem in quantum mechanics. Because time enters quantum mechanics as a parameter rather than a dynamical operator, people have to consider how to construct a time operator and calculate average time whenever they are faced with a time problem related to a physical process. In this paper we present a general investigation on time operator and average time.

Time problem in quantum mechanics and its analysis by the concept of weak measurements

The model of weak measurements is applied to various problems, related to the time problem in quantum mechanics. The review and generalization of the theoretical analysis of the time problem in quantum mechanics based on the concept of weak measurements are presented. A question of the time interval the system spends in the specified state, when the final state of the system is given, is raised. Using the concept of weak measurements the expression for such time is obtained. The results are applied to the tunneling problem. A procedure for the calculation of the asymptotic tunneling and reflection times is proposed. Examples for delta-form and rectangular barrier illustrate the obtained results. Using the concept of weak measurements the arrival time probability distribution is defined by analogy with the classical mechanics. The proposed procedure is suitable to the free particles and to particles subjected to an external potential, as well. It is shown that such an approach imposes an inherent limitation to the accuracy of the arrival time definition.

Time problem in quantum mechanics and weak measurements

A question of the time the system spends in the specified state, when the final state of the system is given, is raised. The model of weak measurements is used to obtain the expression for the time. The conditions for determination of such a time are obtained.