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Mathematics 2016, 4(3), 44; doi:10.3390/math4030044

Exact Discrete Analogs of Canonical Commutation and Uncertainty Relations

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
Academic Editors: Takayuki Miyadera, Paul Busch and Teiko Heinosaari
Received: 3 March 2016 / Revised: 1 June 2016 / Accepted: 24 June 2016 / Published: 28 June 2016
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Abstract

An exact discretization of the canonical commutation and corresponding uncertainty relations are suggested. We prove that the canonical commutation relations of discrete quantum mechanics, which is based on standard finite difference, holds for constant wave functions only. In this paper, we use the recently proposed exact discretization of derivatives, which is based on differences that are represented by infinite series. This new mathematical tool allows us to build sensible discrete quantum mechanics based on the suggested differences and includes the correct canonical commutation and uncertainty relations. View Full-Text
Keywords: quantum theory; canonical commutation relation; uncertainty relations; exact discretization; finite difference quantum theory; canonical commutation relation; uncertainty relations; exact discretization; finite difference
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Tarasov, V.E. Exact Discrete Analogs of Canonical Commutation and Uncertainty Relations. Mathematics 2016, 4, 44.

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