An Introduction to the Inverse Quantum Bound State Problem in One Dimension

TL;DR

This paper presents a method to reconstruct reflectionless quantum potentials in one dimension from finite spectral data, demonstrated on classic quantum systems like the infinite well, harmonic oscillator, and hydrogen atom.

Contribution

It introduces a novel technique for inverse quantum problems, enabling potential reconstruction from limited spectral information in educationally relevant systems.

Findings

Successfully reconstructs potentials for standard quantum systems

Demonstrates applicability to common undergraduate problems

Provides a practical approach for inverse spectral analysis

Abstract

A technique to reconstruct one-dimensional, reflectionless potentials and the associated quantum wave functions starting from a finite number of known energy spectra is discussed. The method is demonstrated using spectra that scale like the lowest energy states of standard problems encountered in the undergraduate curriculum such as: the infinite square well, the simple harmonic oscillator, and the one-dimensional hydrogen atom.