Spherically Symmetric Potentials: Radial Equation

Avoy Jana

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.

Joseph B. Krieger

Anzor Khelashvili, Teimuraz Nadareishvili

There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.

John D. McGervey

Owen L. de Lange, R. E. Raab

Abstract A number of quantum-mechanical problems involve a potential that is spherically symmetric, at least to a first approximation. For such potentials one can consider a ‘onedimensional’ problem, namely the eigenvalue equation for the radial Hamiltonian (see (5.26)), and in this chapter we use operator methods to treat certain important cases for which exact solutions are possible.