Exact Wave Functions of Two-Electron Quantum Rings

Abstract

We demonstrate that the Schrödinger equation for two electrons on a ring, which is the usual paradigm to model quantum rings, is solvable in closed form for particular values of the radius. We show that both polynomial and irrational solutions can be found for any value of the angular momentum and that the singlet and triplet manifolds, which are degenerate, have distinct geometric phases. We also study the nodal structure associated with these two-electron states. Quantum rings are usually modelled by electrons confined to a strict-or quasi-one-dimensional circular space interacting via a short-ranged or Coulomb operator. In this Letter, we focus on the simple system in which two electrons are confined to a ring of radius R and interact via a Coulomb operator. This choice has often been avoided in the literature due to the divergence of the Coulomb interaction at small interelectronic distances.Contrary to frequent claims, systems with two electrons do not inevitably have intractable Schrödinger equations and we show here that, for each electronic state of a two-electron QR, the Schrödinger equation can be solved exactly for a countably infinite set of R values, yielding both polynomial and irrational solutions in terms of the interelectronic distance. Quantum mechanical systems whose Schrödinger equations can be solved in this way, such as the Hooke's law [22] or spherium [23,24] atoms, have ongoing value both for illuminating more complicated systems [25,26] and for testing and developing theoretical approaches, such as DFT [27][28][29][30] and explicitly correlated methods [31].In atomic units ( = m = e = 1), the Hamiltonian of two electrons on a ring of radius R iŝ

Nodal surfaces and interdimensional degeneracies

“…The Pauli hyperplanes are only a subset of the full nodes and it is interesting to note that these nodes are related to the 1D nodes. 24,27,32 The non-interacting nodes are represented in Fig. 1 for a given position of the first electron Ω 1 = (π/6, 0), which is represented by a small yellow sphere.…”

“…This result is not surprising because we know that excited states must have additional nodes in order to be orthogonal to the lowest-energy state, and these additional nodes are usually not imposed by symmetry. 24 …”

Generalized local-density approximation and one-dimensional finite uniform electron gases

“…In Ref [39], we built a GLDA correlation functional ε c [ρ(r), η(r)] based on the "electrons-on-a-ring" paradigm (also called ringium) which corresponds to electrons on a ring interacting through the ring [47][48][49] (see Appendix A).…”

Exact wave functions for concentric two-electron systems

“…In particular, we have shown that one can solve the Schrödinger equation for two electrons confined to a ring [12] or to the surface of a sphere [10,11] for particular values of the ring or sphere radius. In particular, we showed that some of the solutions exhibit the Berry phase phenomenon, i.e.…”

“…In particular, we showed that some of the solutions exhibit the Berry phase phenomenon, i.e. if one of the electrons moves once around the ring and returns to its starting point, the wave function of the system changes sign [12]. Subsequently, Guo et al reported some exact solutions for two vertically coupled quantum rings using the same technique [13] but their analysis is incomplete [14].…”