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Journal of Physics A Mathematical and Theoretical

Comment on ‘Almost-periodic time observables for bound quantum systems’

Eric A. Galapon

Published Nov 19, 2008• 3 citations
Publisher

Abstract

Hall ( 2008 J. Phys. A: Math. Theor. 41 255301 ) makes two claims on a time operator constructed in Galapon (2002 Proc. R. Soc. A 458 2671). We discuss why both claims are wrong.
DOI
10.1088/1751-8113/42/1/018001

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In a recent paper (Hall 2008 J. Phys. A: Math. Gen. 41 255301), I made several critical remarks on a ‘Hermitian time operator’ proposed by Galapon (2002 Proc. R. Soc. A 458 2671). Galapon has correctly pointed out that remarks pertaining to ‘denseness’ of the commutator domain are wrong (Galapon 2008 J. Phys. A: Math. Theor. 42 018001). However, the other remarks still apply, and it is further noted that a given quantum system can be a member of this domain only at a set of times of total measure zero.

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Quantum first time-of-arrival operators

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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.

Journal of Physics A Mathematical and Theoretical•Oct 15, 2008•31 citations
In a recent paper (Hall 2008 J. Phys. A: Math. Gen. 41 255301), I made several critical remarks on a ‘Hermitian time operator’ proposed by Galapon (2002 Proc. R. Soc. A 458 2671). Galapon has correctly pointed out that remarks pertaining to ‘denseness’ of the commutator domain are wrong (Galapon 2008 J. Phys. A: Math. Theor. 42 018001). However, the other remarks still apply, and it is further noted that a given quantum system can be a member of this domain only at a set of times of total measure zero.
Abstract This paper is a rebuttal of the comment by Zhdanov (2018 J. Phys. A: Math. Theor .) with regard to our paper (Russell et al 2017 J. Phys. A: Math. Theor . 50 205302). Parts of the comment either misrepresent the original claims in Russell et al (2017 J. Phys. A: Math. Theor . 50 205302) or consider issues that go beyond the scope of Russell et al (2017 J. Phys. A: Math. Theor . 50 205302). We address all of these points in the rebuttal. A remaining item about the prospect of an accumulation of null sets was already considered in a prior work (Russell et al 2018 J. Phys. A: Math. Theor . 51 335103). Importantly, the comment fails to consider the vast and continually growing set of successful quantum control tests under well-defined conditions, which serves as foundational evidence for the results in Russell et al (2017 J. Phys. A: Math. Theor . 50 205302). In summary, the remarks in the comment either misconstrue the claims of Russell et al (2017 J. Phys. A: Math. Theor . 50 205302) or raise points that are irrelevant to the goal of the original work. Thus, we conclude that the primary claim of the original paper stands: almost all closed finite dimensional quantum systems are trap free given adequate control resources.
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.