Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Quantum states of two particles on concentric spheres

Gregory S. Ezra, R. Stephen Berry

Published Oct 1, 1983• 57 citations
Publisher

Abstract

The model of two particles on a sphere is extended to two particles on concentric spheres (POCS). The quantum states are found for two electrons, one on a sphere of radius 10 a.u. (simulating the shell $n=3$ in He) and the other, on spheres of 10, 15, 25, 50, and 100 a.u. The eigenvalues and densities $\ensuremath{\rho}({\ensuremath{\theta}}_{12})$ exhibit a transition from collective, moleculelike behavior to independent-particle-like behavior with Russell-Saunders coupling. The parallel problem of two particles with electron masses interacting via a repulsive Gaussian potential is also treated and a similar transition from collective to independent-particle behavior found. The principal difference between the two cases is only the region of radius of the larger sphere where the transition occurs.

DOI
10.1103/physreva.28.1989

Related papers

Two interacting electrons in a spherical box: An exact diagonalization study

David C. Thompson, Ali Alavi

We study a system of two electrons interacting with a Coulomb potential in a sphere of radius R, bounded by an infinite wall using exact diagonalization. We have also investigated the influence of an additional parabolic potential (of strength $k)$ arising from a uniform background smeared throughout the sphere. The convergence of the ground state energy of the singlet spin state of the system is investigated as a function of sphere size (essentially ${r}_{s},$ the Wigner--Seitz density parameter) for cases where there is no background potential $(k=0)$ and for when $k\ensuremath{\ne}0.$ With $k=0$ and small ${r}_{s},$ we observe a maximum in the ground state density at the origin of the sphere. At ${r}_{s}\ensuremath{\approx}8\mathrm{a}.\mathrm{u}.,$ the ground state density acquires a minimum at the origin. For this and larger systems we identify the formation of a ``Wigner'' molecule state. We further investigate the ground state density as a function of k and also the correlation hole density as a function of ${r}_{s}$ and k. We invert the Kohn--Sham equation for a two electron system and calculate the local effective potential and correlation potential (to within an additive constant) as functions of the radial coordinate for a number of values of ${r}_{s}$ and k.

Physical review. B, Condensed matter•Dec 31, 2002•40 citations

Electron correlations in quantum dots

D. L. J. Tipton

Quantum dot structures confine electrons in a small region of space. Some properties of semiconductor quantum dots, such as the discrete energy levels and shell filling effects visible in addition spectra, have analogies to those of atoms and indeed dots are sometimes referred to as 'artificial atoms'. However, atoms and dots show some fundamental differences due to electron correlations. For real atoms, the kinetic energy of electrons dominates over their mutual Coulomb repulsion energy and for this reason the independent electron approximation works well. For quantum dots the confining potential may be shallower than that of real atoms leading to lower electron densities and a dominance of mutual Coulomb repulsion over kinetic energy. In this strongly correlated regime the independent electron picture leads to qualitatively incorrect results. This thesis concentrates on few-electron quantum dots in the strongly correlated regime both for quasi-one-dimensional and two-dimensional dots in a square confining potential. In this so-called 'Wigner' regime the ground-state electronic charge density is localised near positions of classical electrostatic minima and the interacting electronic spectrum consists of well separated spin multiplets. In the strongly correlated regime the structure of low-energy multiplets is explained by mapping onto lattice models with extended-Hubbard and Heisenberg effective Hamiltonians. The parameters for these effective models are calculated within a Hartree approximation and are shown to reproduce well the exact results obtained by numerical diagonalisation of the full interacting Hamiltonian. Comparison is made between square dots and quantum rings with full rotational symmetry. In the very low-density regime, direct diagonalisation becomes impractical due to excessive computer time for convergence. In this regime a numerical renormalisation group method is applied to one-dimensional dots, enabling effective spin-interactions to be pr

•Jan 1, 2001•0 citations

Adiabatic connection in density-functional theory: Two electrons on the surface of a sphere

Michael Seidl

Two interacting electrons that are confined to the surface of a sphere have a uniform ground-state (surface) density. The Schr\"odinger equation of this helium-type two-electron system is solved here accurately for different values $\ensuremath{\alpha}∊\mathbb{R}$ of a constant that is multiplied to the electron-electron repulsion ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}}_{\mathit{ee}}$. The correlation structure in the resulting wave functions is analyzed for different values of $\ensuremath{\alpha}$. The asymptotic limits $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ and $\ensuremath{\alpha}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ are treated analytically. Using these results, the ISI (interaction-strength interpolation) model for the density-functional ${E}_{\mathit{xc}}[\ensuremath{\rho}]$ of the exchange-correlation energy in the real system with $\ensuremath{\alpha}=1$ is tested against the exact adiabatic connection in density-functional theory.

Physical Review A•Jun 13, 2007•40 citations

Gravitational self-localization for spherical masses

M. Jääskeläinen

In this work, I consider the center-of-mass wave function for a homogenous sphere under the influence of the self-interaction due to Newtonian gravity. I solve for the ground state numerically and calculate the average radius as a measure of its size. For small masses, $M\ensuremath{\lesssim}{10}^{\ensuremath{-}17}$ kg, the radial size is independent of density, and the ground state extends beyond the extent of the sphere. For masses larger than this, the ground state is contained within the sphere and to a good approximation given by the solution for an effective radial harmonic-oscillator potential. This work thus determines the limits of applicability of the point-mass Newton Schr\"odinger equations for spherical masses. In addition, I calculate the fringe visibility for matter-wave interferometry and find that in the low-mass case, interferometry can in principle be performed, whereas for the latter case, it becomes impossible. Based on this, I discuss this transition as a possible boundary for the quantum-classical crossover, independent of the usually evoked environmental decoherence. The two regimes meet at sphere sizes $R\ensuremath{\approx}{10}^{\ensuremath{-}7}$ m, and the density of the material causes only minor variations in this value.

Physical Review A•Nov 8, 2012•8 citations

Correlation of two weakly attractive particles on a sphere

Robert J. Hinde, R. Stephen Berry

We simulate the behavior of two weakly attractive atoms, either spin-polarized hydrogen atoms or $^{3}\mathrm{He}$ atoms, on a $^{4}\mathrm{He}$ cluster by examining the angular correlation of the atoms as they move on the surface of a sphere. We have calculated converged energy levels, wave functions, and reduced densities ${\mathrm{\ensuremath{\rho}}}^{(\mathit{J})}$(${\mathrm{\ensuremath{\theta}}}_{12}$) for the low-lying states of these two systems. We also give the oscillator strengths for several transitions between low-lying states for the case of spin-polarized hydrogen. The energy levels can be arranged in patterns reminiscent of the rovibrational levels of a linear triatomic molecule, but the reduced densities differ significantly from those predicted by the molecular model. The lowest state for two $^{3}\mathrm{He}$ atoms on a sphere of radius 13.0 bohr is bound, with E=-0.031 90 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$.

Physical Review A•Aug 1, 1990•14 citations

Coherent states for a quantum spin 1/2 particle on a sphere

K. Kowalski, Jakub Rembieliński, Loucas C. Papaloucas

The coherent states for a quantum spin 1/2 particle on a sphere are introduced. These states are marked with points of the classical phase space, that is the position and the angular momentum of a particle.

AIP conference proceedings•Jan 1, 1998•0 citations
The model of two particles on a sphere is extended to two particles on concentric spheres (POCS). The quantum states are found for two electrons, one on a sphere of radius 10 a.u. (simulating the shell n=3n=3n=3 in He) and the other, on spheres of 10, 15, 25, 50, and 100 a.u. The eigenvalues and densities \ensuremathρ(\ensuremathθ12)\ensuremath{\rho}({\ensuremath{\theta}}_{12})\ensuremathρ(\ensuremathθ12​) exhibit a transition from collective, moleculelike behavior to independent-particle-like behavior with Russell-Saunders coupling. The parallel problem of two particles with electron masses interacting via a repulsive Gaussian potential is also treated and a similar transition from collective to independent-particle behavior found. The principal difference between the two cases is only the region of radius of the larger sphere where the transition occurs.
We study a system of two electrons interacting with a Coulomb potential in a sphere of radius R, bounded by an infinite wall using exact diagonalization. We have also investigated the influence of an additional parabolic potential (of strength k)k)k) arising from a uniform background smeared throughout the sphere. The convergence of the ground state energy of the singlet spin state of the system is investigated as a function of sphere size (essentially rs,{r}_{s},rs​, the Wigner--Seitz density parameter) for cases where there is no background potential (k=0)(k=0)(k=0) and for when $k\ensuremath{\ne}0.$ With k=0k=0k=0 and small rs,{r}_{s},rs​, we observe a maximum in the ground state density at the origin of the sphere. At ${r}_{s}\ensuremath{\approx}8\mathrm{a}.\mathrm{u}.,$ the ground state density acquires a minimum at the origin. For this and larger systems we identify the formation of a ``Wigner'' molecule state. We further investigate the ground state density as a function of k and also the correlation hole density as a function of rs{r}_{s}rs​ and k. We invert the Kohn--Sham equation for a two electron system and calculate the local effective potential and correlation potential (to within an additive constant) as functions of the radial coordinate for a number of values of rs{r}_{s}rs​ and k.
Quantum dot structures confine electrons in a small region of space. Some properties of semiconductor quantum dots, such as the discrete energy levels and shell filling effects visible in addition spectra, have analogies to those of atoms and indeed dots are sometimes referred to as 'artificial atoms'. However, atoms and dots show some fundamental differences due to electron correlations. For real atoms, the kinetic energy of electrons dominates over their mutual Coulomb repulsion energy and for this reason the independent electron approximation works well. For quantum dots the confining potential may be shallower than that of real atoms leading to lower electron densities and a dominance of mutual Coulomb repulsion over kinetic energy. In this strongly correlated regime the independent electron picture leads to qualitatively incorrect results. This thesis concentrates on few-electron quantum dots in the strongly correlated regime both for quasi-one-dimensional and two-dimensional dots in a square confining potential. In this so-called 'Wigner' regime the ground-state electronic charge density is localised near positions of classical electrostatic minima and the interacting electronic spectrum consists of well separated spin multiplets. In the strongly correlated regime the structure of low-energy multiplets is explained by mapping onto lattice models with extended-Hubbard and Heisenberg effective Hamiltonians. The parameters for these effective models are calculated within a Hartree approximation and are shown to reproduce well the exact results obtained by numerical diagonalisation of the full interacting Hamiltonian. Comparison is made between square dots and quantum rings with full rotational symmetry. In the very low-density regime, direct diagonalisation becomes impractical due to excessive computer time for convergence. In this regime a numerical renormalisation group method is applied to one-dimensional dots, enabling effective spin-interactions to be pr
Two interacting electrons that are confined to the surface of a sphere have a uniform ground-state (surface) density. The Schr\"odinger equation of this helium-type two-electron system is solved here accurately for different values $\ensuremath{\alpha}∊\mathbb{R}$ of a constant that is multiplied to the electron-electron repulsion ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}}_{\mathit{ee}}$. The correlation structure in the resulting wave functions is analyzed for different values of $\ensuremath{\alpha}$. The asymptotic limits $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ and $\ensuremath{\alpha}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ are treated analytically. Using these results, the ISI (interaction-strength interpolation) model for the density-functional ${E}_{\mathit{xc}}[\ensuremath{\rho}]$ of the exchange-correlation energy in the real system with $\ensuremath{\alpha}=1$ is tested against the exact adiabatic connection in density-functional theory.
In this work, I consider the center-of-mass wave function for a homogenous sphere under the influence of the self-interaction due to Newtonian gravity. I solve for the ground state numerically and calculate the average radius as a measure of its size. For small masses, $M\ensuremath{\lesssim}{10}^{\ensuremath{-}17}$ kg, the radial size is independent of density, and the ground state extends beyond the extent of the sphere. For masses larger than this, the ground state is contained within the sphere and to a good approximation given by the solution for an effective radial harmonic-oscillator potential. This work thus determines the limits of applicability of the point-mass Newton Schr\"odinger equations for spherical masses. In addition, I calculate the fringe visibility for matter-wave interferometry and find that in the low-mass case, interferometry can in principle be performed, whereas for the latter case, it becomes impossible. Based on this, I discuss this transition as a possible boundary for the quantum-classical crossover, independent of the usually evoked environmental decoherence. The two regimes meet at sphere sizes $R\ensuremath{\approx}{10}^{\ensuremath{-}7}$ m, and the density of the material causes only minor variations in this value.
We simulate the behavior of two weakly attractive atoms, either spin-polarized hydrogen atoms or 3He^{3}\mathrm{He}3He atoms, on a 4He^{4}\mathrm{He}4He cluster by examining the angular correlation of the atoms as they move on the surface of a sphere. We have calculated converged energy levels, wave functions, and reduced densities ${\mathrm{\ensuremath{\rho}}}^{(\mathit{J})}$(${\mathrm{\ensuremath{\theta}}}_{12}$) for the low-lying states of these two systems. We also give the oscillator strengths for several transitions between low-lying states for the case of spin-polarized hydrogen. The energy levels can be arranged in patterns reminiscent of the rovibrational levels of a linear triatomic molecule, but the reduced densities differ significantly from those predicted by the molecular model. The lowest state for two 3He^{3}\mathrm{He}3He atoms on a sphere of radius 13.0 bohr is bound, with E=-0.031 90 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$.