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arXiv:quant-ph/9807066 · Physical Review A

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

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

Published Dec 1, 1998• 72 citations
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Abstract

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.

DOI
10.1103/physreva.58.4336

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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.
Previous numerical analyses on the Aharonov-Bohm (AB) operator representing the quantum time-of-arrival (TOA) observable for the free particle have indicated that its eigenfunctions represent quantum states with definite arrival time at the arrival point. In this paper, we give the mathematical proof that this is indeed the case. An essential element of this proof is the consideration of the eigenfunctions of the AB operator with complex eigenvalues. These eigenfunctions can be considered legitimate TOA eigenfunctions because they evolve unitarily to collapse at the arrival point at the time equal to the real part of their eigenvalue. We show that the time-evolved TOA position probability density distribution evaluated at the time equal to the real part of the eigenvalue forms a dirac delta sequence in the limit as the imaginary part of the eigenvalue approaches zero.
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.
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.
We demonstrate that the time operator that measures the time of arrival of a quantum particle into a chosen state can be defined as a self-adjoint quantum-mechanical operator using periodic boundary conditions and applied to wave functions in energy representation. The time becomes quantized into discrete eigenvalues; and the eigenstates of the time operator, i.e., the stroboscopic wave packets introduced recently [Phys. Rev. Lett. 101, 046402 (2008)], form an orthogonal system of states. The formalism provides simple physical interpretation of the time-measurement process and direct construction of normalized, positive definite probability distribution for the quantized values of the arrival time. The average value of the time is equal to the phase time but in general depends on the choice of zero time eigenstate, whereas the uncertainty of the average is related to the traversal time and is independent of this choice. The general formalism is applied to a particle tunneling through a resonant tunneling barrier in one dimension.
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.