Generalized Radial Uncertainty Product for d-Dimensional Hydrogen Atom

TL;DR

This paper extends the radial uncertainty relation to d-dimensional hydrogen atoms, analyzing how quantum uncertainties depend on dimensionality and quantum numbers, providing new insights into higher-dimensional quantum systems.

Contribution

It introduces a generalized radial uncertainty product for d-dimensional hydrogen atoms, deriving analytical expressions and exploring the impact of dimensionality on quantum uncertainties.

Findings

Uncertainty product depends on quantum numbers and dimension d

Derived explicit formulas for radial wave functions and uncertainties

Revealed the role of dimensionality in quantum uncertainty relations

Abstract

This paper presents a comprehensive analysis of the generalized radial uncertainty product for the d-dimensional non-relativistic Hydrogen atom in position space. Utilizing the framework of quantum mechanics in d-dimensional spherical coordinates, the study extends the standard radial uncertainty relation to higher dimensions. Taking the solution of the radial Schrodinger equation, the normalized radial wave functions, expectation values, and uncertainties in both position and momentum space are rigorously evaluated. The analytical derivations reveal the dependence of the uncertainty product on the principal and angular quantum numbers, as well as the dimensional parameter d. The results provide deeper insight into the role of dimensionality in quantum uncertainty relations and their implications for higher-dimensional quantum systems