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arXiv:quant-ph/0409006 · Lithuanian Journal of Physics

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

Julius Ruseckas, Bronislovas Kaulakys

Published Jan 1, 2004• 0 citations
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Abstract

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 tunnelling problem.A procedure for the calculation of the asymptotic tunnelling and reflection times is proposed.Examples for δ-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.

DOI
10.3952/lithjphys.44301

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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 tunnelling problem.A procedure for the calculation of the asymptotic tunnelling and reflection times is proposed.Examples for δ-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.
We consider a number of aspects of the problem of defining time observables in quantum theory. Time observables are interesting quantities in quantum theory because they often cannot be associated with self-adjoint operators. Their definition therefore touches on foundational issues in quantum theory. Various operational approaches to defining time observables have been proposed in the past. Two of the most common are those based on pulsed measurements in the form of strings of projection operators and continuous measurements in the form of complex potentials. One of the major achievements of this thesis is to prove that these two operational approaches are equivalent. However operational approaches are somewhat unsatisfying by themselves. To provide a definition of time observables which is not linked to a particular measurement scheme we employ the decoherent, or consistent, histories approach to quantum theory. We focus on the arrival time, one particular example of a time observable, and we use the relationship between pulsed and continuous measurements to relate the decoherent histories approach to one based on complex potentials. This lets us compute the arrival time probability distribution in decoherent histories and we show that it agrees with semiclassical expectations in the right limit. We do this both for a free particle and for a particle coupled to an environment. Finally, we consider how the results discussed in this thesis relate to those derived by coupling a particle to a model clock. We show that for a general class of clock models the probabilities thus measured can be simply related to the ideal ones computed via decoherent histories.
The theoretical analysis of time observables in quantum mechanics has been controversial and is still subject to debate. In the present work, a measurement-based approach to quantum arrival times is studied in detail. It is based on the emission time of the first spontaneous photon from a a two-level atom moving into a laser-illuminated region. This quantum-optical model is shown to give an operational understanding of various ideal proposals for quantum arrival-time distributions as well as for local kinetic energy densities. As a main result, a first measurement-based procedure to Kijowski`s arrival-time distribution is given and the approach is used to generalize this distribution to the case of particles interacting with an external potential.
We formulate quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles passing through a barrier at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time of arrival determined in C. Anastopoulos and N. Savvidou, J. Math. Phys. 47, 122106 (2006). This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that for a class of localized initial states, the detection probability allows for an identification of tunneling time with the classic phase time. We also establish limits to the definability of tunneling time. We then generalize these results to a sequential measurement setup: the phase-space properties of the particles are determined by an unsharp sampling before their attempt to cross the barrier. For such measurements the tunneling time is defined as a genuine observable. This allows us to construct a probability distribution for its values that is definable for all initial states and potentials. We also identify a regime in which these probabilities correspond to a tunneling-time operator.
It is argued that the time of arrival cannot be precisely defined and measured in quantum mechanics. By constructing explicit toy models of a measurement, we show that for a free particle it cannot be measured more accurately then $\ensuremath{\Delta}{t}_{A}\ensuremath{\sim}{1/E}_{k},$ where Ek{E}_{k}Ek​ is the initial kinetic energy of the particle. With a better accuracy, particles reflect off the measuring device, and the resulting probability distribution becomes distorted. It is shown that a time-of-arrival operator cannot exist, and that approximate time-of-arrival operators do not correspond to the measurements considered here.
In this paper 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 the Schrödinger’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.