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Communications in Theoretical Physics

Study Hankel Transforms and Properties of Bessel Function via Entangled State Representation Transformation in Quantum Mechanics

Fan Hong‐Yi, Yong Wang

Published May 1, 2006• 3 citations
Publisher

Abstract

In Phys. Lett. A 313 (2003) 343 we have found that the self-reciprocal Hankel transformation (HT) is embodied in quantum mechanics by a transform between two entangled state representations of continuum variables. In this work we study Hankel transforms and properties of Bessel function via entangled state representations' transformation in quantum mechanics.

DOI
10.1088/0253-6102/45/5/012

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Study Hankel Transforms and Properties of Bessel Function via Entangled State Representation Transformation in Quantum Mechanics

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在 Phys。Lett。一 313 (2003 ) 343 我们发现了自我相互的 Hankeltransformation (HT ) 被在连续统变量的二个纠缠的州的代表之间的变换在量力学表达。在这个工作，我们在量力学经由纠缠的州的表示的转变学习 Hankel 变换和性质 ofBessel 功能。

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Communications in Theoretical Physics•Jan 1, 2006•4 citations
In Phys. Lett. A 313 (2003) 343 we have found that the self-reciprocal Hankel transformation (HT) is embodied in quantum mechanics by a transform between two entangled state representations of continuum variables. In this work we study Hankel transforms and properties of Bessel function via entangled state representations' transformation in quantum mechanics.
Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between [Formula: see text] and [Formula: see text], two induced entangled state representations, and working with them, we derive fractional Fourier–Hankel transformation (FrFHT) caused by the operator [Formula: see text], where [Formula: see text] is named the core operator and is essential to the fractional transformation. The fractional property (additive rule) of the FrFHT can be explicitly proved.
Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between | q , r 〉 and ( s , r ′|, two induced entangled state representations are given, and working with them we derive fractional squeezing-Hankel transform (FrSHT) caused by the operator e − i α ( a 1 † a 2 † + a 1 a 2 ) e − i π a 2 † a 2 , which is an entangled fractional squeezing transform operator. The additive rule of the FrSHT can be explicitly proved.