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Application of moment wavelet transform to quantum mechanics

Asmerom Ogbazghi

Published Jul 1, 1998• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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 .
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.
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.