A new model for the quantum mechanics of the Hydrogen atom

Abstract

In this paper we introduce a new model for the quantum-mechanical system of the hydrogen atom. We start with a four-dimensional Lorentzian quadratic space $(V,q)$ and let $C \subset V$ be the corresponding cone. The Hilbert space of our model, denoted by $H$, consists of $L^2$ functions on the cone, and observables are represented by operators in the algebra $D(C)$ of algebraic differential operators on $C$. We introduce a distinguished Schwartz subspace $H^{\infty}$ of $H$ that is naturally a $D(C)$-module. The Schr\"{o}dinger operator in our system is represented by a Schr\"{o}dinger family of operators in $D(C)$. We compute the spectrum of the Schr\"{o}dinger family in the Schwartz space $H^{\infty}$ and show that it coincides with the spectrum in physics, and that solutions in $H^{\infty}$ correspond to the usual solutions in physics. The main differences from the standard model are as follows. First, we use the cone $C$ instead of $\mathbb{R}^3$ as our configuration space. As a result, the group of geometric symmetries of our configuration space is $O(q)\simeq O(3,1)$ rather than $O(3)\ltimes \mathbb{R}^3$. Second, we use only algebraic operators with no singularities. Third, we do not impose any specific boundary conditions on solutions of our equations; these are all encoded in the Schwartz space $H^{\infty}$.