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arXiv:quant-ph/0512096 · Physical Review A

Confined quantum time of arrival for the vanishing potential

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

Published Dec 6, 2005• 38 citations
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Abstract

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.

DOI
10.1103/physreva.72.062107

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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.
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.
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.
It has always been believed that no self-adjoint and canonical time of arrival operator can be constructed within the confines of standard quantum mechanics. In this Letter we demonstrate the otherwise. We do so by pointing out that there is no a priori reason in demanding that canonical pairs form a system of imprimitivities. We then proceed to show that a class of self-adjoint and canonical time of arrival (TOA) operators can be constructed for a spatially confined free particle. And then discuss the relatiobship between the non-self-adjointess of the TOA operator for the unconfined particle and the self-adjointness of the confined one.
The unexpected connection is unravelled between the collapse of the wave function on the appearance of particle and the quantum time-of-arrival problem in one dimension. To do so, a theory of quantum first time of arrival is developed in the interacting case for arbitrary arrival point in one dimension based on a self-adjoint and canonical coarse graining of a time-of-arrival operator that derives the classical time-of-arrival observable. The appearance of particle in quantum mechanics is then considered in the light of this theory. It is found that the appearance of particle arises as a combination of the collapse of the initial wave function into one of the eigenfunctions of the time-of-arrival operator, followed by a unitary Schrödinger evolution of the eigenfunction.