Information-Theoretic Properties of the Half-Line Coulomb Potential

TL;DR

This paper explores the information-theoretic properties of the half-line Coulomb potential, revealing Fisher length as a key measure of uncertainty and its relation to wavefunction nodes and energy levels.

Contribution

It introduces the Fisher length as an effective uncertainty measure and analyzes its relationship with wavefunction nodes and energy in the Coulomb system.

Findings

Fisher length is the proper measure of uncertainty in the system.

Position Fisher length is proportional to the number of wavefunction nodes.

Position Fisher length follows a square-root energy law.

Abstract

The half-line one-dimensional Coulomb potential is possibly the simplest D-dimensional model with physical solutions which has been proved to be successful to describe the behaviour of Rydberg atoms in external fields and the dynamics of surface-state electrons in liquid helium, with potential applications in constructing analog quantum computers and other fields. Here, we investigate the spreading and uncertaintylike properties for the ground and excited states of this system by means of the logarithmic measure and the information-theoretic lengths of Renyi, Shannon and Fisher types; so, far beyond the Heisenberg measure. In particular, the Fisher length (which is a local quantity of internal disorder) is shown to be the proper measure of uncertainty for our system in both position and momentum spaces. Moreover the position Fisher length of a given physical state turns out to be not only directly proportional to the number of nodes of its associated wavefunction, but also it follows a square-root energy law.