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International Journal of Modern Physics A

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

Eric A. Galapon

Published Dec 20, 2006• 29 citations

Abstract

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.

DOI: 10.1142/s0217751x06034215

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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 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.

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.

This paper undertakes the problem of constructing first time-of-arrival operators for arbitrary everywhere analytic potentials in the space of operators spanned by the Bender–Dunne basis (Bender and Dunne 1989 Phys. Rev. D 40 3504). The operators are conjugate with their respective system Hamiltonians and reduce to their corresponding classical time-of-arrival expressions for infinitesimal ℏ. The supraquantized time-of-arrival operators (Galapon 2004 J. Math. Phys. 45 3180) and the confined time-of-arrival operators (Galapon et al 2004 Phys. Rev. Lett. 93 180406, Galapon 2006 J. Mod. Phys. A 21 6351) are then shown to be two distinct representations of the time-of-arrival operators constructed in the Bender–Dunne space.