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arXiv:2503.10227 · Physical review. D/Physical review. D.

Hydrogen atom in a fuzzy spherical cavity

Matej Hrmo, Samuel Kováčik, Patrik Rusnák, Juraj Tekel

Published Mar 6, 2026• 0 citations
arXivHTMLPDF

Abstract

The fuzzy onion model formed by connecting a set of concentric fuzzy spheres of increasing radius is motivated by studies of quantum space but can also be used to study standard physics. The main feature of the model is that functions in three-dimensional space—like scalar fields or wave functions—are expressed in terms of Hermitian matrices of a certain structure. Relevant equations are then matrix equations, and some problems, such as searching for the energy spectrum for fixed quantum numbers ( l , m ) , can be expressed as an eigenvalue problem. We show how this simple approach can reproduce the results of other studies analyzing the hydrogen atom in a spherical cavity. We also test the effect of the short-distance quantum structure of the space on these solutions—not looking for the phenomenological consequences, as the scale of quantum space is many orders below the order of the Bohr radius, but to understand the effect of quantum space in general. We observe a set of solutions without a classical counterpart which have been suggested also in a former theoretical study.

DOI
10.1103/9ycw-jkws

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The fuzzy onion model formed by connecting a set of concentric fuzzy spheres of increasing radius is motivated by studies of quantum space but can also be used to study standard physics. The main feature of the model is that functions in three-dimensional space—like scalar fields or wave functions—are expressed in terms of Hermitian matrices of a certain structure. Relevant equations are then matrix equations, and some problems, such as searching for the energy spectrum for fixed quantum numbers ( l , m ) , can be expressed as an eigenvalue problem. We show how this simple approach can reproduce the results of other studies analyzing the hydrogen atom in a spherical cavity. We also test the effect of the short-distance quantum structure of the space on these solutions—not looking for the phenomenological consequences, as the scale of quantum space is many orders below the order of the Bohr radius, but to understand the effect of quantum space in general. We observe a set of solutions without a classical counterpart which have been suggested also in a former theoretical study.
The fuzzy onion model is a three-dimensional quantum space model expressed in terms of Hermitian matrices of a certain form. It is a convenient model as it is manageable and can be easily put on a computer. It can also be related to various physical contexts, capturing the essence of the quantum structure of space. In this work, we briefly summarise the construction of the model and then show how it can be used for the effects of quantum space on spherically symmetrical problems, the Coulomb one, for example. A possible connection to the earlier stages of the universe is discussed.
The exact solution to the problem of a hydrogen atom confined in a spherical well (CHA) is discussed; the standard results for the unconfined hydrogen atom (UHA) are recovered as the sphere size becomes large compared to the Bohr radius. The solutions are characterized by a set of three quantum numbers N (= 1, 2, 3,…), L (= 0, 1, 2,…), and M (= − L, − L + 1,…, L − 1, L), and the energy eigenvalues, in contrast to the situation in the UHA, depend on both N and L. All members of a given family n = N + L, however, evolve asymptotically toward the same energy level in the large-sphere limit, recovering the typical n2 degeneracy of the UHA. Besides numerically exact solutions for arbitrary sphere sizes, rigorous analytical approximations are provided for the physically relevant strong- and weak-confinement regimes. A conjecture concerning the ordering of the energy levels is rigorously confirmed. The validity of the virial theorem, Kato's cusp condition, and the role played by the density as an alternative basic variable for the case of the CHA are discussed.
There are certain prime features of physical systems—notably rotation and its associated angular momentum—that simply do not appear in a one-dimensional world. This chapter introduces the three-dimensional Schrodinger equation and applies it to the analysis of a particle in a three-dimensional box and to the spherically symmetric wave functions of hydrogen. The time-independent Schrodinger equation can be viewed in a way that gives it a family resemblance to many other differential equations of physics, and which expresses its structure in very general terms. A computer solution uses dimensionless variables based on natural units. The Schrodinger theory does not have the limitations of the Bohr theory. Schrodinger theory incorporates the well-established wave nature of matter and can be generalized to analyze systems for which the Bohr theory has proven inadequate: multi-electron atoms, chemical properties of atoms, molecular and solid-state physics and chemistry, as well as more subtle effects in hydrogen itself.
Certain one- and two-dimensional reductions of the three-dimensional Schrödinger equation of the hydrogen atom are considered. These reductions are carried out from the point of view of the two common sets of space coordinates: Cartesian and spherical. The resulting systems have features that relate more readily to the old quantum theory models of Bohr and Sommerfeld than the general three-dimensional hydrogen atom. Furthermore, the considerations yield interesting insights into the quantum mechanics of the hydrogen atom and may serve as helpful intermediary preparation, in an introductory presentation of the subject, for the unreduced three-dimensional case.
The incompatibility between two observables in quantum theory is described by Heisenberg's uncertainty principle. In this work, we study spatial confinement effects on Heisenberg's uncertainty principle for a hydrogen atom located at the center of an impenetrable spherical cavity with radius ro{r}_{o}ro​. Both the radial and vector representation of the uncertainty principle are considered. For this, we solve the Schr\"odinger equation numerically within a finite-differences approach. We find that for small cavity sizes the values of $\mathrm{\ensuremath{\Delta}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{r}\mathrm{\ensuremath{\Delta}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{r}$ (radial) bunch according to the number of nodes and that for Rydberg states, i.e., large excitation, they become more coherent, satisfying exactly Heisenberg's uncertainty principle, in contrast to the vector description. However, for the vector case, we find that $\mathrm{\ensuremath{\Delta}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{r}}$ degenerates for small cavity sizes and bunches according to the principal quantum number nnn for large cavities. We find that the behavior of $\mathrm{\ensuremath{\Delta}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$ is responsible for the breaking of the energy degeneracy for confined quantum systems. This occurs when the confinement radius is of the order of the orbital size, as determined by the electron average distance $\ensuremath{\langle}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{r}\ensuremath{\rangle}$. In addition, we estimate the critical cavity size for which relativistic effects become relevant and verify that the relativistic corrections to the energy, obtained from first-order perturbation theory, become important when the total energy of the atom surpasses 93.845 Hartree, corresponding to 10% of the speed of light, which is fulfilled for cavity sizes ro<1{r}_{o}<1ro​<1 a.u.