Radial Uncertainty Product for Spherically Symmetric Potential in Position Space

TL;DR

This paper derives and analyzes the radial uncertainty relation for quantum systems with spherically symmetric potentials, exploring its implications for key models like hydrogen, spherical well, and harmonic oscillator.

Contribution

It introduces a radial uncertainty product analogous to Cartesian uncertainty, providing analytical and numerical insights for various spherically symmetric quantum systems.

Findings

Radial uncertainty product depends on quantum numbers and potential parameters.

Analytical expressions for radial uncertainties are derived for key potentials.

Numerical results illustrate the behavior of the uncertainty product across different states.

Abstract

This paper presents a detailed analysis of the radial uncertainty product for quantum systems with spherically symmetric potentials. Using the principles of quantum mechanics, the study derives the radial uncertainty relation analogous to the Cartesian form and investigates its implications for three key spherically symmetric potentials: the Hydrogen atom, the infinite spherical potential well, and the spherical harmonic oscillator, all within the non-relativistic regime. Employing the Schrodinger equation in spherical coordinates, the paper rigorously evaluates the normalized radial wave functions, expectation values, and uncertainties associated with both position and momentum. Analytical derivations and numerical computations highlight the dependence of the uncertainty product on quantum numbers and system-specific parameters.