Virial Theorem and Hypervirial Theorem in a spherical geometry

TL;DR

This paper extends the Virial and Hypervirial Theorems to spherical geometries in classical and quantum mechanics, deriving new relations and a perturbation theorem applicable without wave functions, demonstrated through harmonic oscillator and Coulomb system examples.

Contribution

It introduces the Hypervirial relations in spherical spaces and develops a wave-function-free perturbation theorem using the Hellmann-Feynman approach.

Findings

Derived Hypervirial relations in spherical geometries.

Formulated a perturbation theorem without wave functions.

Illustrated methods with harmonic oscillator and Coulomb systems.

Abstract

The Virial Theorem in the one- and two-dimensional spherical geometry are presented, in both classical and quantum mechanics. Choosing a special class of Hypervirial operators, the quantum Hypervirial relations in the spherical spaces are obtained. With the aid of the Hellmann-Feynman Theorem, these relations can be used to formulate a \emph{perturbation theorem without wave functions}, corresponding to the Hypervirial-Hellmann-Feynman Theorem perturbation theorem of Euclidean geometry. The one-dimensional harmonic oscillator and two-dimensional Coulomb system in the spherical spaces are given as two sample examples to illustrate the perturbation method.