A Novel Approach to Schrödinger's Wave Equation: Utilizing Metric Tensor in Spherical Coordinates

Abstract

In quantum mechanics, the Schrödinger equation is fundamental for describing particle wave functions, traditionally within flat spacetime, ignoring gravitational effects. This paper introduces the Howusu Metric Tensor to extend the Schrödinger equation into spherical coordinates, accommodating gravitational fields that are regular and continuous with a reciprocal decrease at infinity. This leads to the derivation of the Riemannian Schrödinger equation, offering insights into quantum behavior in curved spacetime. Building on previous work integrating quantum mechanics with general relativity and Finsler geometry, our approach addresses the limitations in capturing gravitational subtleties. By incorporating the Howusu Metric Tensor, our model accounts for gravitational potential in spherical coordinates, providing a more precise description of quantum phenomena under gravity. The resulting Riemannian Schrödinger equation reveals new quantum behavior influenced by gravitational forces, opening new research possibilities in cosmology and astrophysics, where quantum-gravitational interactions are key. This study demonstrates the advantages of using the Howusu Metric Tensor over previous models, highlighting its potential to unify quantum mechanics with gravitational effects more coherently and comprehensively.