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Journal of Physics Conference Series

Energy spectrum of the Coulomb problem in the fuzzy onion space

Samuel Kováčik

Published Dec 1, 2025• 0 citations
Publisher

Abstract

Abstract We describe a model of three-dimensional quantum space. Starting from a well-studied model of the fuzzy sphere, we extend it by connecting a large — potentially infinite — collection of such spheres to represent a three-dimensional volume. We verify that the chosen Hamiltonian is appropriate; in the continuum limit, the spectrum converges to that of standard quantum mechanics. Specifically, we analyse the behaviour of the spectrum of the Coulomb problem in the finite case.

DOI
10.1088/1742-6596/3152/1/012012

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Abstract We describe a model of three-dimensional quantum space. Starting from a well-studied model of the fuzzy sphere, we extend it by connecting a large — potentially infinite — collection of such spheres to represent a three-dimensional volume. We verify that the chosen Hamiltonian is appropriate; in the continuum limit, the spectrum converges to that of standard quantum mechanics. Specifically, we analyse the behaviour of the spectrum of the Coulomb problem in the finite case.
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 procedure commonly used in textbooks for determining the eigenvalues and eigenstates for a particle in an attractive Coulomb potential is not symmetric in the way the boundary conditions at r = 0 and r → ∞ are considered. We highlight this fact by solving a model for the Coulomb potential with a cutoff (representing the finite extent of the nucleus). In the limit that the cutoff is reduced to zero, we recover the standard result, albeit in a non-standard way. This example is used to emphasize that a more consistent approach to solving the Coulomb problem in quantum mechanics requires an examination of the non-standard solution. The end result is, of course, the same.
The authors discuss exactly solvable Schrodinger Hamiltonians corresponding to a surface delta interaction supported by a sphere and various generalisations thereof. First they treat the pure delta sphere model; self-adjointness of the Hamiltonian, spectral properties, stationary scattering theory, approximation by scaled short-range Hamiltonians. Next they extend the model by adding a point interaction at the centre of the sphere or, alternatively, a Coulomb interaction. Finally the whole analysis is extended to the case of a delta ' sphere interaction, first taken alone, then superimposed on a point interaction or a Coulomb potential.
Frequency sum rules are derived in extended quantum systems of non relativistic fermions from a minimal set of assumptions on dynamics in infinite volume, for ground and thermal states invariant under space translations or a lattice subgroup. For the jellium Coulomb model, they imply the one point result for the plasmon energy spectrum in the zero momentum limit. In general, the density waves energy spectrum is shown to converge, in the limit of large wavelenght, to a point measure at zero frequency, for any number of fermion fields and potentials with integrable second derivatives. For low momentum, ~ k^2 for potentials V with r^2 d_i d_j V integrable, ~ k^{a-d+2} for potentials decaying at infinity as 1/r^a, d-2 < a < d, d the space dimensions. For one component models with short range interactions, the fourth momentum of the frequency is expressed, at lowest order in k, purely in terms of the three point correlation function of the density.