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Acta Physica Sinica

Wavelet transform of odd- and even-binomial states

Jun Song, Ye‐Jun Xu, Fan Hong‐Yi

Published Jan 1, 2011• 5 citations
Publisher

Abstract

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.

DOI
10.7498/aps.60.084208

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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.
Abstract By using the technique of integral within an ordered product of operators, we recast classical wavelet transform to a matrix element of the squeezing‐displacing operator U (μ, s ) between the mother wavelet vector 〈ψ| and the state vector | f 〉 to be transformed, i.e., we propose that 〈ψ| U (μ, s )| f 〉 can be considered as a new kind of spectrum for analyzing the quantum state | f 〉. In this way, we propose the wavelet‐ transform spectrum for quantum chemical states. As an example, we carry out the numerical calculation of wavelet‐transform spectrum for the binomial state. It seems to us that this kind of spectrum can be used to recognize a variety of quantum chemical states. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012
In the context of quantum mechanics we employ the technique of integration within an ordered product of operators to recast the classical wavelet transform to a squeezing-displacing transform between the mother wavelet vector and the state vector to be transformed. In this way we propose the wavelet-transform spectrum for quantum optical states. For some typical states we obtain numerical results which imply that the spectrum can be used to recognize a variety of quantum optical states, and the inverse wavelet transform has the possibility to play a role in quantum state engineering.
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.