The energy balancing parameter

TL;DR

The paper introduces an energy balancing parameter method to estimate relationships among variables in equilibrium systems with incomplete potential energy knowledge, applicable to classical and quantum cases.

Contribution

It presents a new parameter method that simplifies energy analysis in equilibrium systems, bypassing complex quantum calculations and providing estimates for previously unsolved problems.

Findings

Successfully applied to hydrogen-like atoms and elastic potentials

Provides estimates for cases with incomplete potential energy data

Simplifies quantum mechanical calculations using elementary algebra

Abstract

A parameter method is introduced in order to estimate the relationship among the various variables of a system in equilibrium, where the potential energy functions are incompletely known or the quantum mechanical calculations very difficult. No formal proof of the method is given; instead, a sufficient number of valuable examples are shown to make the case for the method's usefulness in classical and quantum systems. The mathematical methods required are quite elementary: basic algebra and minimization of power functions. This method blends advantageously with a simple but powerful approximate method for quantum mechanics, sidestepping entirely formal operators and differential equations. It is applied to the derivation of various well-known results involving centrally symmetric potentials for a quantum particle such as the hydrogen-like atom, the elastic potential and other cases of interest. The same formulas provide estimates for previously unsolved cases. PACS: 03.65.-w 30.00.00