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Journal of Modern Optics

Wavelet transform in the context of quantum mechanics and new orthogonal property of mother wavelets in parameter space

Liyun Hu, Hong-Yi Fan

Published Jun 20, 2008• 13 citations
Publisher

Abstract

Based on the quantum mechanical version of wavelet transform (WT) W ψ f(μ,s) = ⟨ψ|U(μ,s)| f ⟩ [Fan, H.Y.; Lu, H.L. Opt. Lett. 2006, 31, 407] we further prove Parseval theorem and inversion formula for WT in the context of quantum mechanics. The concise proof not only manifestly shows the merit of Dirac's representation theory, but also leads to a new orthogonal property of mother wavelets in parameter space.

DOI
10.1080/09500340701823177

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Based on the quantum mechanical version of wavelet transform (WT) W ψ f(μ,s) = ⟨ψ|U(μ,s)| f ⟩ [Fan, H.Y.; Lu, H.L. Opt. Lett. 2006, 31, 407] we further prove Parseval theorem and inversion formula for WT in the context of quantum mechanics. The concise proof not only manifestly shows the merit of Dirac's representation theory, but also leads to a new orthogonal property of mother wavelets in parameter space.
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.