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Journal of Physics A Mathematical and General

Continuous wavelet transform analysis of quantum systems with rational potentials

Carlos R. Handy, Romain Murenzi

Published Jul 7, 1997• 9 citations
Publisher

Abstract

Given a one-dimensional Sturm - Liouville Schrödinger problem with rational polynomial potential, we can generate the continuous wavelet transform (CWT) for its discrete states, thereby permitting the systematic multiscale reconstruction of the corresponding bound-state wavefunction. A key component in this is the use of properly dilated (a) and translated (b) moments, , which readily transform the configuration space Hamiltonian into a finite set of dynamically coupled, linear, first-order differential equations in the dilation-related variable, : The infinite scale problem is readily solved through moment quantization methods and used to generate the moments at all scales. We demonstrate the essentials through the rational fraction potential, , and the Coulomb potential.

DOI
10.1088/0305-4470/30/13/022

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Given a one-dimensional Sturm - Liouville Schrödinger problem with rational polynomial potential, we can generate the continuous wavelet transform (CWT) for its discrete states, thereby permitting the systematic multiscale reconstruction of the corresponding bound-state wavefunction. A key component in this is the use of properly dilated (a) and translated (b) moments, , which readily transform the configuration space Hamiltonian into a finite set of dynamically coupled, linear, first-order differential equations in the dilation-related variable, : The infinite scale problem is readily solved through moment quantization methods and used to generate the moments at all scales. We demonstrate the essentials through the rational fraction potential, , and the Coulomb potential.
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.
Over the last decade, Handy and Bessis have developed a moment-problem-based, multiscale quantization theory, the eigenvalue moment method (EMM), which has proven effective in solving singular, strongly coupled, multidimensional Schr\"odinger Hamiltonians. We extend the scope of EMM by demonstrating its essential role in the generation of wavelet transforms for one-dimensional quantum systems. Combining this with the function-wavelet reconstruction formulas currently available, we are able to recover the wave function systematically, from first principles, through a multiscale process proceeding from large spatial scales to smaller ones. This accomplishment also addresses another outstanding problem, that of reconstructing a function from its moments. For the class of problems considered, the combined EMM-wavelet analysis yields a definitive solution. \textcopyright{} 1996 The American Physical Society.
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 .
The nondecimated or translation-invariant wavelet transform (NDWT) is a central tool in classical multiscale signal analysis, valued for its stability, redundancy, and shift invariance. This paper develops two complementary quantum formulations of the NDWT that embed these classical properties coherently into quantum computation. The first formulation is based on the epsilon-decimated interpretation of the NDWT and realizes all circularly shifted wavelet transforms simultaneously by promoting the shift index to a quantum register and applying controlled circular shifts followed by a wavelet analysis unitary. The resulting construction yields an explicit, fully unitary quantum representation of redundant wavelet coefficients and supports coherent postprocessing, including quantum shrinkage via ancilla-driven completely positive trace preserving maps. The second formulation is based on the Hadamard test and uses diagonal phase operators to probe scale-shift wavelet structure through interference, providing direct access to shift-invariant energy scalograms and multiscale spectra without explicit coefficient reconstruction. Together, these two approaches demonstrate that redundancy and translation invariance can be exploited rather than avoided in the quantum setting. Applications to denoising, feature extraction, and spectral scaling illustrate how quantum NDWTs provide a flexible and physically meaningful foundation for multiscale quantum signal processing.
For one-dimensional Schrödinger quantum systems, the correlation expression S ( x , τ, a ) ≡ Ψ*( x − τ/2/ a )Ψ( x + τ/2) satisfies a fourth-order linear differential equation with regard to x . This generalizes the result previously derived by Handy (2001 J. Phys. A: Math. Gen. 34 L271), and Handy and Wang (2001 J. Phys. A: Math. Gen. 34 8297), with regard to S ( x , 0, 1). We are then able to incorporate this within a generalized Wigner transform representation, through the use of scalets (Handy C R and Brooks H A 2001 J. Phys. A: Math. Gen. 34 3577). Energy quantization is achieved through a moment problem positivity analysis (focusing on the moments of the probability density, ∣Ψ∣ 2 ) at a = 1, τ = 0. The wavefunction, Ψ, is then generated through a multiscale analysis proceeding from the unity scale, extended into the zero scale limit. The power moments, μ p = ∫ d x x p Ψ( x ), can be generated through a similar procedure. We present the general formalism and apply it to the (i x ) 3 potential.