The Energy Eigenvalue for the Singular Wave Function of the Three Dimensional Dirac Delta Schrodinger Potential via Distributionally Generalized Quantum Mechanics

TL;DR

This paper rigorously determines the bound state energy for a 3D Dirac delta potential in quantum mechanics using a distributional approach, avoiding renormalization and confirming results consistent with existing literature.

Contribution

It introduces a distributionally generalized Schrödinger framework to solve the 3D Dirac delta potential problem without renormalization, providing a mathematically rigorous solution.

Findings

Bound state energy computed in a well-posed distributional setting

Results agree with existing mathematical and physics literature

Established a distributional version of the Hellmann-Feynman theorem

Abstract

Unlike the situation for the 1d Dirac delta derivative Schrodinger pseudo potential (SPP) and the 2d Dirac delta SPP, where the indeterminacy originates from a lack of scale in the first and both a lack of scale as well as the wave function not being well defined at the support of the generalized function SPP; the obstruction in 3d Euclidean space for the Schrodinger equation with the Dirac delta as a SPP only comes from the wave function (the $L^2$ bound sate solution) being singular at the compact point support of the Dirac delta function (measure). The problem is solved here in a completely mathematically rigorous manner with no recourse to renormalization nor regularization. The method involves a distributionally generalized version of the Schrodinger theory as developed by the author, which regards the formal symbol "$H\psi$" as an element of the space of distributions, the topological dual vector space to the space of smooth functions with compact support. Two main facts come to light. The first is the bound state energy of such a system can be calculated in a well-posed context, the value of which agrees with both the mathematical and theoretical physics literature. The second is that there is then a rigorous distributional version of the Hellmann-Feynman theorem.