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Communications in Theoretical Physics

Parseval Theory of Complex Wavelet Transform for Wavelet Family Including Rotational Parameters

Nanrun Zhou, Fang Jia

Published Sep 1, 2008• 2 citations
Publisher

Abstract

A rotational parameter Rθ has been introduced to complex wavelet transform (CWT). The rotational CWT (RCWT) corresponds to a matrix element ψ|U2(θ;μ;k)|F in the context of quantum mechanics, where U2(θ;μ;k) is a two-mode rotational displacing-squeezing operator in the η| representation. Based on this, the Parseval theorem and the inversion formula of RCWT have been proved. The concise proof not only manifestly shows the merit of Dirac's representation theory but also leads to a new orthogonal property of complex mother wavelets in parameter space.

DOI
10.1088/0253-6102/50/3/12

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A rotational parameter Rθ has been introduced to complex wavelet transform (CWT). The rotational CWT (RCWT) corresponds to a matrix element ψ|U2(θ;μ;k)|F in the context of quantum mechanics, where U2(θ;μ;k) is a two-mode rotational displacing-squeezing operator in the η| representation. Based on this, the Parseval theorem and the inversion formula of RCWT have been proved. The concise proof not only manifestly shows the merit of Dirac's representation theory but also leads to a new orthogonal property of complex mother wavelets in parameter space.
For the efficient identification of quantum states, we propose the notion of linear canonical wavelet transform in the framework of quantum mechanics. Using the machinery of Dirac representation theory and integration within an ordered product of operators, we recast the linear canonical wavelet transform to a matrix element of the squeezing–displacing operator U(μ,s)KM between analyzing vector ⟨ψ| and two-mode quantum state vector |f⟩ to be transformed. We also derive the inner product relation and inversion formula for the linear canonical wavelet transform in the realm of quantum mechanics. Lastly, we present an explicit example for the lucid implementation of linear canonical wavelet transform in identifying the quantum states.
Based on the quantum mechanics representation, we propose a novel 3D entangled wavelet transformation. This kind of three entangled state representation can be generated theoretically by using a two-cascaded beam splitter, which has the completeness and orthogonal property. Some properties of 3D WTs are discussed, including the Parseval theory of 3D WTs, its inverse formula, and the orthogonality of the mother wavelets in parameter space. In addition, the mother wavelet condition for the 3D WTs is also derived.
In the context of quantum mechanics the classical wavelet transform of a function f with the mother wavelet can be recast into a matrix element of the squeezing-displacing operator U(,s) as 〈|U(,s) |f〉. The technique of integral within an ordered product of operators is used to support this theory. Based on this, wavelet transforms are done for even- and odd-binomial states, and the corresponding numerical calculation leads to the spectrum of wavelet transform, which is helpful for recognizing the difference between even- and odd-states.
The symplectic wavelet transform, which is related to the quantum optical Fresnel transform, is developed to the symplectic-dilation mixed wavelet transform (SDWT). The SDWT involves both a real-variable dilation-transform and a complex-variable symplectic transform and possesses well-behaved properties such as the Parseval theorem and the inversion formula. The entangled-coherent state representation not only underlies the SDWT but also helps to derive the corresponding quantum transform operator whose counterpart in classical optics is the lens-Fresnel mixed transform.