Coordinate Space Modification of Fock's Theory-Harmonic Tensors in the Quantum Coulomb Problem

TL;DR

This paper modifies Fock's theory of the hydrogen atom by shifting from momentum space to coordinate space, using tensor methods to simplify the Schrödinger equation into a 4D Laplace form and deriving solutions including the Stark effect.

Contribution

It introduces a coordinate space modification of Fock's theory, employing invariant tensor methods to transform and solve the hydrogen atom problem in 4D space.

Findings

Derived a 4D Laplace equation form of the Schrödinger equation

Obtained solutions for eigenfunctions in the modified framework

Calculated the quadratic Stark effect explicitly

Abstract

We consider Fock's fundamental theory of the hydrogen atom in momentum space which allows a realization of the previously predicted rotation group of a three-dimensional (3D) sphere in four-dimensional (4D) space. We then modify Fock's theory and abandon the momentum space description. To transform and simplify the theory, we use invariant tensor methods of electrostatics in 3D and 4D spaces. We find a coordinate 4D space where the Schrodinger equation becomes the 4D Laplace equation. The transition from harmonic 4D polynomials to original 3D physical space is algebraic and involves derivatives with respect to a coordinate that is interpreted as time. We obtain a differential equation for eigenfunctions in the momentum space and find its solutions. A concise calculation of the quadratic Stark effect is given. The Schwinger resolvent is derived by the method of harmonic polynomials. Vector ladder operators are also considered.