Morlet wavelets in quantum mechanics

TL;DR

This paper explores the application of Morlet wavelets in quantum mechanics, highlighting their advantages in localized analysis and providing explicit forms for inverse and four-dimensional transforms.

Contribution

It introduces explicit forms for inverse and four-dimensional Morlet wavelet transforms tailored for quantum mechanics applications.

Findings

Morlet wavelets are effective for localized quantum analysis

Explicit inverse Morlet transform is derived

Covariant four-dimensional Morlet wavelet form provided

Abstract

Wavelets offer significant advantages for the analysis of problems in quantum mechanics. Because wavelets are localized in both time and frequency they avoid certain subtle but potentially fatal conceptual errors that can result from the use of plane wave or delta function decomposition. Morlet wavelets are particularly well-suited for this work: as Gaussians, they have a simple analytic form and they work well with Feynman path integrals. To take full advantage of Morlet wavelets we need an explicit form for the inverse Morlet transform and a manifestly covariant form for the four-dimensional Morlet wavelet. We supply both here.