In a previous paper [V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997)] we introduced a self-adjoint operator $\mathcal{T}^(X)$ 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 ${J}^{(+)}(X)$ , 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 ${J}^{(+)}(X)$ 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.In a previous paper [V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997)] we introduced a self-adjoint operator $\mathcal{T}^(X)$ 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 ${J}^{(+)}(X)$, 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 ${J}^{(+)}(X)$ 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.

arXiv:quant-ph/9709037Physical Review AFeb 1, 199834 citations

Arrival time from the general theory of quantum time distributions

Tajron Jurić, Hrvoje Nikolić

arXiv:2107.08777The European Physical Journal PlusMay 1, 20228 citations

Quantum arrival time for open systems

James M. Yearsley

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 the Lindblad form. We specialize to the case of quantum Brownian motion, and show that after a time of order the localization time of the current becomes positive. We show that the arrival time probabilities can then be written in terms of a positive operator-valued measure (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 localization time, but that there is a fundamental limitation on the accuracy $\ensuremath{\delta}t$, with which they can be specified which obeys $E\ensuremath{\delta}t\ensuremath{\gg}\ensuremath{\hbar}$. 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.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 the Lindblad form. We specialize to the case of quantum Brownian motion, and show that after a time of order the localization time of the current becomes positive. We show that the arrival time probabilities can then be written in terms of a positive operator-valued measure (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 localization time, but that there is a fundamental limitation on the accuracy $\ensuremath{\delta}t$, with which they can be specified which obeys $E\ensuremath{\delta}t\ensuremath{\gg}\ensuremath{\hbar}$. 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.

arXiv:1003.6079Physical Review AJul 30, 201018 citations

The time distribution of quantum events

Danijel Jurman, Hrvoje Nikolić

arXiv:2010.07575Physics Letters AFeb 24, 202113 citations

On the “standard” quantum mechanical approach to times of arrival

C. Richard Leavens

Physics Letters AOct 1, 200235 citations

In a previous paper [V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997)] we introduced a self-adjoint operator

\mathcal{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)

, 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)

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.