Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Canadian Journal of Physics

Time of Arrival Operator

Mohsen Razavy

Published Dec 15, 1971• 17 citations
Publisher

Abstract

The problem of finding proper quantal operators for the time of arrival, when the source is at infinite distance from the detector, is investigated. It is shown that at least two Hermitian operators can be constructed which, in the Hilbert space of positive energy spectrum, are self-adjoint. The spectra of these operators are continuous and their eigenfunctions form an orthonormal set.

DOI
10.1139/p71-367

Related papers

Arrival time in quantum mechanics

V. Delgado, Juan Gonzalo Muga

A self-adjoint operator with dimensions of time is explicitly constructed, and it is shown that its complete and orthonormal set of eigenstates can be used to define consistently a probability distribution of the time of arrival at a spatial point.

arXiv:quant-ph/9704010•Physical Review A•Nov 1, 1997•124 citations

THEORY OF QUANTUM FIRST TIME OF ARRIVAL VIA SPATIAL CONFINEMENT I: CONFINED TIME OF ARRIVAL OPERATORS FOR CONTINUOUS POTENTIALS

Eric A. Galapon

The constructions of two classes of self-adjoint and compact first time of arrival operators for confined systems under arbitrary, everywhere continuous potential are detailed, extending in the interacting case the concept of confined quantum time of arrival operators first developed for the free particle. One class is the quantized confined time of arrival operators and another is the class of algebra preserving confined time of arrival operators. The former is the projection of the quantization of the classical time of arrival, and is constructed by solving the quantization problem of the multiple-valued and non-everywhere-real-valued classical time of arrival for arbitrary potential. The later arises to address the nonconjugacy with the Hamiltonian of the entire class of the quantized confined time of arrival operators, and is constructed by solving the obstruction to quantization present in Euclidean space. These two sets of operators coincide for linear systems; but differ for nonlinear systems, with the former as the leading term of the latter. The confined time of arrival operators for potentials representative of linear and nonlinear systems are numerically investigated and demonstrated to have the same dynamical behaviors as those of the free confined time of arrival operators. In particular, the eigenfunctions evolve according to Schrödinger equation such that the corresponding probabilities of locating the quantum particle in the neighborhood of the arrival point are maximum at their respective eigenvalues.

International Journal of Modern Physics A•Dec 20, 2006•29 citations

Time-of-arrival formalism for the relativistic particle

J León

A suitable operator for the time-of-arrival at a detector is defined for the free relativistic particle in (3 + 1) dimensions. For each detector position there exists a subspace of detected states in the Hilbert space of solutions to the Klein - Gordon equation. Orthogonality and completeness of the eigenfunctions of the time-of-arrival operator apply inside this subspace, opening up a standard probabilistic interpretation.

arXiv:quant-ph/9608013•Journal of Physics A Mathematical and General•Jul 7, 1997•57 citations

Space-time properties of free-motion time-of-arrival eigenfunctions

Juan Gonzalo Muga, C. Richard Leavens, José P. Palao

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.

arXiv:quant-ph/9807066•Physical Review A•Dec 1, 1998•72 citations

Confined quantum time of arrival for the vanishing potential

Eric A. Galapon, Roland Cristopher F. Caballar, Ricardo T. Bahague

We give full account of our recent report in E. A. Galapon, R. Caballar, and R. Bahague, Phys. Rev. Lett. 93, 180406 (2004), where it is shown that formulating the free quantum time of arrival problem in a segment of the real line suggests rephrasing the quantum time of arrival problem to finding a complete set of states that evolve to unitarily arrive at a given point at a definite time. For a spatially confined particle, here it is shown explicitly that the problem admits a solution in the form of an eigenvalue problem of a class of compact and self-adjoint time of arrival operators derived by a quantization of the classical time of arrival. The eigenfunctions of these operators are numerically demonstrated to unitarily arrive at the origin at their respective eigenvalues.

arXiv:quant-ph/0512096•Physical Review A•Dec 6, 2005•38 citations

Confined Quantum Time of Arrivals

Eric A. Galapon, Roland Cristopher F. Caballar, Ricardo T. Bahague

We show that formulating the quantum time of arrival problem in a segment of the real line suggests rephrasing the quantum time of arrival problem to finding states that evolve to unitarily collapse at a given point at a definite time. For the spatially confined particle, we show that the problem admits a solution in the form of an eigenvalue problem of a compact and self-adjoint time of arrival operator derived by a quantization of the classical time of arrival, which is canonically conjugate with the Hamiltonian in a closed subspace of the Hilbert space.

arXiv:quant-ph/0302036•Physical Review Letters•Oct 29, 2004•86 citations
The problem of finding proper quantal operators for the time of arrival, when the source is at infinite distance from the detector, is investigated. It is shown that at least two Hermitian operators can be constructed which, in the Hilbert space of positive energy spectrum, are self-adjoint. The spectra of these operators are continuous and their eigenfunctions form an orthonormal set.
The constructions of two classes of self-adjoint and compact first time of arrival operators for confined systems under arbitrary, everywhere continuous potential are detailed, extending in the interacting case the concept of confined quantum time of arrival operators first developed for the free particle. One class is the quantized confined time of arrival operators and another is the class of algebra preserving confined time of arrival operators. The former is the projection of the quantization of the classical time of arrival, and is constructed by solving the quantization problem of the multiple-valued and non-everywhere-real-valued classical time of arrival for arbitrary potential. The later arises to address the nonconjugacy with the Hamiltonian of the entire class of the quantized confined time of arrival operators, and is constructed by solving the obstruction to quantization present in Euclidean space. These two sets of operators coincide for linear systems; but differ for nonlinear systems, with the former as the leading term of the latter. The confined time of arrival operators for potentials representative of linear and nonlinear systems are numerically investigated and demonstrated to have the same dynamical behaviors as those of the free confined time of arrival operators. In particular, the eigenfunctions evolve according to Schrödinger equation such that the corresponding probabilities of locating the quantum particle in the neighborhood of the arrival point are maximum at their respective eigenvalues.
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.
We give full account of our recent report in E. A. Galapon, R. Caballar, and R. Bahague, Phys. Rev. Lett. 93, 180406 (2004), where it is shown that formulating the free quantum time of arrival problem in a segment of the real line suggests rephrasing the quantum time of arrival problem to finding a complete set of states that evolve to unitarily arrive at a given point at a definite time. For a spatially confined particle, here it is shown explicitly that the problem admits a solution in the form of an eigenvalue problem of a class of compact and self-adjoint time of arrival operators derived by a quantization of the classical time of arrival. The eigenfunctions of these operators are numerically demonstrated to unitarily arrive at the origin at their respective eigenvalues.
We show that formulating the quantum time of arrival problem in a segment of the real line suggests rephrasing the quantum time of arrival problem to finding states that evolve to unitarily collapse at a given point at a definite time. For the spatially confined particle, we show that the problem admits a solution in the form of an eigenvalue problem of a compact and self-adjoint time of arrival operator derived by a quantization of the classical time of arrival, which is canonically conjugate with the Hamiltonian in a closed subspace of the Hilbert space.