Quantum Mechanics in Wavelet Basis

TL;DR

This paper introduces a multi-scale wavelet basis method for quantum mechanics, enabling intuitive analysis of quantum fluctuations at different resolutions and simplifying solutions for systems with natural length scales.

Contribution

It presents a novel wavelet-based framework for quantum problems, allowing natural interpretation and efficient approximation through matrix diagonalization.

Findings

Wavelet basis provides a natural interpretation of quantum fluctuations.

The method simplifies solving quantum systems with natural length scales.

Application to the harmonic oscillator demonstrates effectiveness.

Abstract

We describe a multi-scale resolution approach to analyzing problems in Quantum Mechanics using Daubechies wavelet basis. The expansion of the wavefunction of the quantum system in this basis allows a natural interpretation of each basis function as a quantum fluctuation of a specific resolution at a particular location. The Hamiltonian matrix constructed in this basis describes couplings between different length scales and thus allows for intuitive volume and resolution truncation. In quantum mechanical problems with a natural length scale, one can get approximate solution of the problem through simple matrix diagonalization. We illustrate this approach using the example of the standard quantum mechanical simple harmonic oscillator.