On integrals associated with the Free particle wave packet

TL;DR

This paper explores properties of integrals related to free particle wave packets in quantum mechanics and derives formulas for half-integer arguments of the Riemann zeta function, connecting quantum physics and number theory.

Contribution

It provides new insights into integrals associated with free particle wave packets and derives formulas for specific values of the Riemann zeta function, linking quantum physics and mathematics.

Findings

Analysis of integrals related to wave packets

Formulas for half-integer Riemann zeta function values

Discussion of literature on related integrals

Abstract

We discuss some properties of integrals associated with the free particle wave packet, $\psi(x,t),$ which are solutions to the time-dependent Schr$\ddot{o}$dinger equation for a free particle in one dimension. Some noteworthy discussion is made in relation to integrals which have appeared in the literature. We also obtain formulas for half-integer arguments of the Riemann zeta function.