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arXiv · 2603.14969

A new model for the quantum mechanics of the Hydrogen atom

Joseph Bernstein·Eyal Subag
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Abstract

To every Lorentzian quadratic space $(V,q)$ of even dimension $n$ such that $n\geq 4$ we attach a canonical algebraic quantum mechanical model for a corresponding generalized hydrogen atom system. In our model the configuration space is the regular null cone $C$ of the quadratic space. The Hilbert space $H$ is a canonical $L^2$ space on the cone $C$, and observables are realized in the algebra $D(C)$ of algebraic differential operators on $C$. We also construct a distinguished Schwartz space $S(H)\subset H$, which carries a self-adjoint action of $D(C)$ and encodes the boundary conditions of the standard theory. The role of the Schr\"odinger operator is played by a one-parameter Schr\"odinger family of operators in $D(C)$. We explain how the model relates to the realization of the minimal representation of $O(n,2)$ on $H$. For $n=4$, which corresponds to the physical hydrogen atom system, we prove that the spectrum of the Schr\"odinger family on the upper-half component $S(H)_+$ coincides with the usual spectrum of the hydrogen atom and that the corresponding solution spaces recover the standard physical solutions. The spectrum of the Schr\"odinger family on the lower-half component $S(H)_-$ gives additional positive-energy solution spaces not present in the usual formulation.

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Joseph Bernstein, Eyal Subag. 2026-03-16. A new model for the quantum mechanics of the Hydrogen atom. https://arxiv.org/abs/2603.14969

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