An exact lower energy bound for the infinite square well potential

TL;DR

This paper presents an exact lower energy bound for a quantum particle in an infinite square well, derived through a variational approach independent of the Schrödinger equation, illustrating a pedagogical example for quantum mechanics education.

Contribution

It introduces a novel variational method to determine an exact lower energy bound for the infinite square well, independent of the Schrödinger equation, suitable for undergraduate teaching.

Findings

Derived an exact lower energy bound for the infinite square well

Identified the wavefunction that achieves the bound

Provided a pedagogical example of variational methods in quantum mechanics

Abstract

We give a lower bound for the energy of a quantum particle in the infinite square well. We show that the bound is exact and identify the well-known element that fulfils the equality. Our approach is not directly dependent on the Schroedinger equation and illustrates an example where the wavefunction is obtained directly by energy minimization. The derivation presented can serve as an example of a variational method in an undergraduate level university course in quantum mechanics.