Operator Space Manifold Theory: Modeling Quantum Operators with a Riemannian Manifold

Abstract

AbstractThe Half-Transform Ansatz (HTA) is a proposed method to solve hyper-geometric equations in Quantum Phase Space by transforming a differential operator to an algebraic variable and including a specific exponential factor in the wave function, but the mechanism which provides this solution scheme is not known. Analysis of the HTA’s application to the Hydrogen atom suggests an underlying mechanism which the HTA is a part of. Observations of exponential factors that act on the wave function naturally suggest modeling quantum operator definitions as a point on a Riemannian manifold in the 4D Operator Space, a novel idea we call the Operator Space Manifold Theory. On this manifold, we explore the concepts of superposition, regions of unique energy eigenvalues, and translation operators. We also find the theoretical backing to derive the HTA and how Operator Space Manifold Theory can be used to describe and solve quantum systems by manipulating how a quantum state perceives position and momentum.