Time-of-arrival probabilities and quantum measurements

Abstract

We study the construction of probability densities for time-of-arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time-of-arrival appear naturally when one considers histories. The definition of time-of-arrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of Schr\"odinger's equation does not naturally define a probability density at the continuum limit, but also because the procedure one follows is sensitive on the interpretation of the reduction procedure. We consider the issue in Copenhagen quantum mechanics and in history-based schemes like consistent histories. The benefit of the latter is that it allows a proper passage to the continuous limit--there are however problems related to the quantum Zeno effect and decoherence. We finally employ the histories-based description to construct Positive-Operator-Valued-Measures (POVMs) for the time-of-arrival, which are valid for a general Hamiltonian. These POVMs typically depend on the resolution of the measurement device; for a free particle, however, this dependence cancels in the physically relevant regime and the POVM coincides with that of Kijowski.Comment: 40 pages, 1 fig.--expanded argumentation, version to appear in JM

Time-of-arrival probabilities and quantum measurements. III. Decay of unstable states

“…It is important to emphasize that by construction p(t) is linear with respect to the initial density matrix, positive-definite, normalized (when the alternative of non-detection is also taken into account) and a genuine density with respect to time. We refer the reader to [1] for the physical assumptions relevant to the definition of this POVM and to [2] for its detailed construction for the case relevant to tunneling.…”

“…To solve this problem, we use the result of [1], namely the construction of a POVM for the time-of-arrival for particles for a generic HamiltonianĤ. This POVM provides a unique determination of the probability distribution p(t) for the time-of-arrival.…”

“…This is a third in a series of papers [1,2], which studies the properties and applications of a Positive-Operator-Valued-Measure for the time-of-arrival. In [1] this POVM was constructed and it was applied to the free particle case, where it coincided with the one of Kijowski [3].…”

“…In [1] this POVM was constructed and it was applied to the free particle case, where it coincided with the one of Kijowski [3]. In [2], this POVM was adapted to the tunneling case: it led the determination of tunneling time and through a generalization to sequential measurements it provided a probability distribution for the times of arrival.…”

“…For this purpose, we use a Positive-Operator-Valued-Measure (POVM) for the time-of-arrival determined in [1]. This only depends on the initial state, the Hamiltonian and the location of the detector.…”