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arXiv:quant-ph/9905023 · Physical Review A

Free-motion time-of-arrival operator and probability distribution

I. L. Egusquiza, Juan Gonzalo Muga

Published Dec 9, 1999• 85 citations
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Abstract

We reappraise and clarify the contradictory statements found in the literature concerning the time-of-arrival operator introduced by Aharonov and Bohm in Phys. Rev. 122, 1649 (1961). We use Naimark's dilation theorem to reproduce the generalized decomposition of unity (or positive-operator-valued measures) from any self-adjoint extension of the operator, emphasizing a natural one, which arises from the analogy with the momentum operator on the half-line. General time operators are set within a unifying perspective. It is shown that they are not in general related to the time of arrival, even though they may have the same form.

DOI
10.1103/physreva.61.012104

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We reappraise and clarify the contradictory statements found in the literature concerning the time-of-arrival operator introduced by Aharonov and Bohm in Phys. Rev. 122, 1649 (1961). We use Naimark's dilation theorem to reproduce the generalized decomposition of unity (or positive-operator-valued measures) from any self-adjoint extension of the operator, emphasizing a natural one, which arises from the analogy with the momentum operator on the half-line. General time operators are set within a unifying perspective. It is shown that they are not in general related to the time of arrival, even though they may have the same form.
The properties of the time-of-arrival operator for free motion introduced by Aharonov and Bohm [Phys. Rev. 122, 1649 (1961)] and of its self-adjoint variants are studied. The domains of applicability of the different approaches are clarified. It is shown that the arrival time of the eigenfunctions is not sharply defined. However, strongly peaked real-space (normalized) wave packets constructed with narrow Gaussian envelopes centered on one of the eigenfunctions provide an arbitrarily sharp arrival time.
In a previous paper [V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997)] we introduced a self-adjoint operator T(X)\mathcal{T}^(X)T(X) whose eigenstates can be used to define consistently a probability distribution of the time of arrival at a given spatial point. In the present work we show that the probability distribution previously proposed can be well understood on classical grounds in the sense that it is given by the expectation value of a certain positive-definite operator J(+)(X){J}^{(+)}(X)J(+)(X), which is nothing but a straightforward quantum version of the modulus of the classical current. For quantum states highly localized in momentum space about a certain momentum ${p}_{0}\ensuremath{\ne}0$, the expectation value of J(+)(X){J}^{(+)}(X)J(+)(X) becomes indistinguishable from the quantum probability current. This fact may provide a justification for the common practice of using the latter quantity as a probability distribution of arrival times.
A realization of the concept of ``crossing state'' invoked, but not implemented, by Wigner, allows advancement in two important aspects of the formalization of the time of arrival in quantum mechanics: (i) For free motion, we find that the limitations described by Aharonov et al. in Phys. Rev. A 57, 4130 (1998) for the time-of-arrival uncertainty at low energies for certain measurement models are in fact already present in the intrinsic time-of-arrival distribution of Kijowski ${\ensuremath{\Pi}}_{K};$ (ii) we have also found a covariant generalization of this distribution for arbitrary potentials and positions.
Using the orthodox Weyl -- Wigner -- Stratonovich -- Cohen (WWSC) quantization rule we construct a time -- of -- arrival operator for a free particle on the circle. It is shown that this operator is self -- adjoint but the careful analysis of its properties suggests that it cannot represent a `physical' time -- of -- arrival observable. The problem of a time -- of -- arrival observable for the `waiting screen' is also considered. A method of avoiding the quantum Zeno effect is proposed and the positive operator valued measure (POV -- measure) or the generalized positive operator valued measure (GPOV -- measure) describing quantum time -- of -- arrival observable for the waiting screen are found.
A relativistic version of the Aharonov-Bohm time-of-arrival operator for spin-0 particles was constructed by Razavi [Il Nuovo Cimento B 63, 271 (1969)]. We study the operator in detail by taking its rigged Hilbert space extension. It is shown that the rigged Hilbert space extension of the operator provides more insights into the time-of-arrival problem that goes beyond Razavi's original results. This allows us to use time-of-arrival eigenfunctions that exhibit unitary arrival to construct time-of-arrival distributions. The expectation value is also calculated and shown that particles can arrive earlier or later than expected classically. Last, the constructed time-of-arrival distribution and expectation value are shown to be consistent with special relativity.
We revisit the arguments underlying two well-known arrival-time distributions in quantum mechanics, viz., the Aharonov–Bohm–Kijowski (ABK) distribution, applicable for freely moving particles, and the quantum flux (QF) distribution. An inconsistency in the original axiomatic derivation of Kijowski’s result is pointed out, along with an inescapable consequence of the ‘negative arrival times’ inherent to this proposal (and generalizations thereof). The ABK free-particle restriction is lifted in a discussion of an explicit arrival-time set-up featuring a charged particle moving in a constant magnetic field. A natural generalization of the ABK distribution is in this case shown to be critically gauge-dependent. A direct comparison to the QF distribution, which does not exhibit this flaw, is drawn (its acknowledged drawback concerning the quantum backflow effect notwithstanding).