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Journal of Physics A Mathematical and General

Moment quantization and (A-adic) discrete-continuous wavelet transform theory

Carlos R. Handy, Romain Murenzi

Published Nov 8, 1999• 8 citations
Publisher

Abstract

We establish how moment quantization directly leads to a complete ( A -adic) discrete-continuous wavelet transform theory in contrast to the approximate reconstruction methods published earlier (1996 Phys. Rev. A 54 3754, 1997 J. Phys. A: Math. Gen. 30 4709).
DOI
10.1088/0305-4470/32/46/309

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The space of polynomials maps onto itself under affine transformations, . This suggests that a moment reformulation of continuous wavelet transform (CWT) theory (the affine convolution, , of a signal, or wavefunction, ) should lead to significant simplifications in its implementation. We present a comprehensive formalism, with numerical examples, that inextricably links moment quantization (MQ) and CWT theory. For rational fraction potential problems and mother wavelets of the form ( Q ( x ) an appropriate polynomial), MQ permits a more efficient and accurate (in a pointwise convergent sense) CWT implementation; whereas, CWT broadens the scope of applicability for MQ methods, and is its natural extension when a more global approximation is desired. Our formalism also gives one justification for the empirical superiority manifested by previous MQ studies, as compared with dyadic wavelet reconstruction methods. We implement our formalism in the context of the quartic, sextic and octic anharmonic oscillator potentials, and demonstrate the flexibility of the method by treating both the Mexican hat wavelet transform, as well as that based on the mother wavelet .
Wavelet transform theory is an efficient multiscale formalism for analysing local structures. This philosophy, when incorporated within quantum mechanics, demands that there be a naturally corresponding, localized quantization theory, in contrast to the variational formalisms in the literature. Through the recently established equivalency formalism between moment quantization theory and continuous wavelet transform theory (Handy C R and Murenzi R 1998 J. Phys. A: Math. Gen. 31 9897 and Handy C R and Murenzi R 1999 J. Phys. A: Math. Gen. 32 8111), we argue that a new quantization prescription can be defined in which the kinetic energy term is set to zero at the (complex) turning points (or turning hypersurfaces). We establish this, both for one- and two-dimensional systems, and clarify the relevancy of multiscale wavelet analysis in this quantization process.
In this work we reconstruct Quantum Mechanical wave functions for some confining potentials, using the moment-wavelet method of Handy and Murenzi. This method consists in transforming the Schrodinger equation into an equivalent continuous wavelet transform (CWT) representation involving scaled and translated moments, µ1/ α,b (p)=∫χpe-Q\\( Ψ(x +b), where ϐ -Q becomes the mother wavelet. The discrete bound states are determined through a multiscale process involving the integration of a finite number of coupled linear first order differential equation in the moments µ1/ α,b (p). The underlying initial value problem depends on moment quantization methods to determine the infinite scale (a = ∞) moments and energy. Using this method we calculate the energies and wavefunctions for the first two quantum states of quartic and dectic anharmonic oscillator potentials, V(x) = mχ2 + gχ4, V(χ) = χ 2+χ10 respectively.
We present a general method for calculating relevant moments of the wavelet transform.Explicit results are given for the time, frequency, and scale moments.Using the results obtained, we show that the wavelet transform has unique characteristics that are not possessed by other methods.We discuss whether these unusual characteristics are physically or mathematically important.Exactly solvable examples are given, and the results are contrasted to those of the standard methods such as the spectrogram and the Wigner distribution.