Generalization of Quantum Mechanical Wave Equation in Spherical Coordinate Using Great Metric Tensors and a Variable Gravitational Scalar Potential

In this research work, the Riemannian Laplacian operator for a spherical system which varies with time, radial distance and time was obtained using the great metric tensors and a varying gravitational scalar potential. Furthermore the obtained Laplacian operator was used to obtain the generalized quantum mechanical wave equation for particles within this field. The Laplacian operator obtained in this work reduces to the well known Laplacian operator in the limit of Capture_11.PNG, and it contained post Euclid or pure Riemannian correction terms of all orders of Capture_2.PNG . Also the generalized quantum mechanical wave equation obtained, in the limit of Capture_12.PNGreduces to the well known Schrodinger mechanical wave equation, and in the limit of Capture_21.PNGcontained additional correction terms not found in the well known Schrodinger wave equation. Hence the results in this work satisfy the Principle of Equivalence in Physics.