Time as an observable in nonrelativistic quantum mechanics

Abstract

The nonrelativistic Schrödinger equation for motion of a structureless particle in four-dimensional space-time entails a well-known expression for the conserved four-vector field of local probability density and current that are associated with a quantum state solution to the equation. Under the physical assumption that each spatial, as well as the temporal, component of this current is observable, the position in time becomes an operator and an observable in that the weighted average value of the time of the particle's crossing of a complete hyperplane can be simply defined: the theory predicts, and experiment is presumed to be able to observe, the integral over the hyperplane of the normal component of probability current, weighted by the time coordinate. In conventional formulations the hyperplane is always spacelike, i.e., is a time=constant hyperplane in Galilean relativity, and the result is then trivial. A nontrivial result is obtained if the plane is not of this type. When the space-time coordinates are (t, x, y, z), the paper analyzes in detail the case that the hyperplane is of the type z=constant. Particles can cross such a hyperplane in either direction, so it proves convenient to introduce an indefinite metric, and correspondingly a sesquilinear inner product with non-Hilbert space structure, for the space of quantum states on such a surface. Since the metric is indefinite, an uncertainty principle involving the dispersion of the crossing time and the dispersion of its conjugate momentum does not appear to be derivable from the theory. A detailed formalism for computing average crossing times on a z=constant hyperplane, and average dwell times and delay times for a zone of interaction contained between a pair of z=constant hyperplanes, is presented.

New approach to the quantum tunneling process: Characteristic times for transmission and reflection

“…Apart from the SWPA to deal with the CM of a wave packet, in the same or different setting the tunneling problem, a variety of alternative approaches (see reviews [1,9,10,11,12,13] and references therein) to introduce various characteristic times for a tunneling particle have also been developed. Among the alternative concepts, of interest are that of the dwell time [14,15,16,17], that of the Larmor time [18,19,20,21,22] to give the way of measuring the dwell time, and the concept of the time of arrival which is based on introducing either a suitable time operator (see, for example, [23,24,25,26,27]) or the positive operator valued measure (see review [12]). Besides, of importance are the studies of the temporal aspects of tunneling on the basis of the Feynman, Bohmian and Wigner approaches to deal with the random trajectories of particles (see, for example, [28,29,30,31,32] and references therein).…”

Quantum Mechanical Operator of Time

“…Since this objection, there have been many attempts to address time and/or its operator in QM. The extensive lists of references one can find in [1][2], while the present approach have some concrete similarities with those given in [3][4][5][6][7].…”

Vibrational and rotational energy transfer in collisions of vibrationally excited HF molecules with Ar atoms

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