# Evolution of piecewise polynomial wave functions

## TL;DR

This paper studies how polynomial-based wave functions evolve over time in quantum mechanics, providing exact solutions for free particles and oscillators using Fresnel integrals, and discusses extensions to higher dimensions.

## Contribution

It introduces a method to exactly compute the evolution of spline wave functions, including complex cases and higher dimensions, expanding analytical tools in quantum wave dynamics.

## Key findings

- Exact evolution formulas for spline wave functions using Fresnel integrals
- Extension of methods to two and three dimensions
- Analysis of the immediate spatial extension of compact wave functions

## Abstract

For a non-relativistic particle, we consider the evolution of wave functions that consist of polynomial segments, usually joined smoothly together. These spline wave functions are compact (that is, they are initially zero outside a finite region), but they immediately extend over all available space as they evolve. The simplest splines are the square and triangular wave functions in one dimension, but very complicated splines have been used in physics. In general the evolution of such spline wave functions can be expressed in terms of antiderivatives of the propagator; in the case of a free particle or an oscillator, all the evolutions are expressed exactly in terms of Fresnel integrals. Some extensions of these methods to two and three dimensions are discussed.