A new model for the quantum mechanics of the Hydrogen atom
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914470513606656 |
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| author | Bernstein, Joseph Subag, Eyal |
| author_facet | Bernstein, Joseph Subag, Eyal |
| contents | 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 Schrödinger operator in our system is represented by a Schrödinger family of operators in $D(C)$.
We compute the spectrum of the Schrödinger 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 $H^{\infty}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14969 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A new model for the quantum mechanics of the Hydrogen atom Bernstein, Joseph Subag, Eyal Mathematical Physics Algebraic Geometry Representation Theory 8110, 16S32, 22E47, 47L60, 81Q99 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 Schrödinger operator in our system is represented by a Schrödinger family of operators in $D(C)$. We compute the spectrum of the Schrödinger 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 $H^{\infty}$. |
| title | A new model for the quantum mechanics of the Hydrogen atom |
| topic | Mathematical Physics Algebraic Geometry Representation Theory 8110, 16S32, 22E47, 47L60, 81Q99 |
| url | https://arxiv.org/abs/2603.14969 |