Using quantum mechanics for calculation of different infinite sums

TL;DR

This paper presents a novel method using quantum mechanics to analytically evaluate certain infinite sums, linking quantum problems to mathematical functions like the Riemann zeta, and offers educational insights.

Contribution

It introduces a new approach connecting quantum mechanics with the calculation of infinite sums, expanding analytical tools for mathematical physics.

Findings

Calculated Riemann zeta function values using quantum mechanical mean energy.

Demonstrated the method's applicability to various exactly solvable quantum problems.

Provided educational insights into superposition principle through the method.

Abstract

We demonstrate that certain class of infinite sums can be calculated analytically starting from a specific quantum mechanical problem and using principles of quantum mechanics. For simplicity we illustrate the method by exploring the problem of a particle in a box. Twofold calculation of the mean value of energy for the polynomial wave function inside the well yields even argument $p$ ($p>2$) of Riemann zeta and related functions. This method can be applied to a wide class of exactly solvable quantum mechanical problems which may lead to different infinite sums. Besides, the analysis performed here provides deeper understanding of superposition principle and presents useful exercise for physics students.