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arXiv:1907.02024

From wave-functions to single electron densities

Ugo Bindini, Luigi De Pascale

Published Jul 3, 2019• 0 citations
arXivHTMLPDF

Abstract

We investigate some of the properties of the mapping from wave-functions to single particle densities, partially answering an open question posed by E. H. Lieb in 1983.
DOI
10.48550/arxiv.1907.02024

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Abstract An attempt is made to substantiate the absence of wave properties for a single electron. What is commonly called the corpuscular-wave duality or the wave properties of matter, which appear in the form of a diffraction pattern on the target of certain kind, is a result of the obligatory involvement of a large number of electrons in the diffraction process. In the context of quantum mechanics, a form of the diffraction pattern can be predicted using the rule of adding probability amplitudes, which, in the simplest case, are solutions to the Schrödinger equation for free particles. Therefore, the answer to the question posed in the title sounds something like this: a single electron has no wave properties; however, they become apparent whenever there are many electrons and a suitable target.
The optimized single-particle wave functions contained in the parameters of the Hubbard model (t and U) were determined for an infinite atomic chain. In effect, the electronic properties of the chain as a function of interatomic distance R were obtained and compared for the Lieb - Wu exact solution (LW), the Gutzwiller-Wave-Function approximation (GWF), and the Gutzwiller-ansatz case (GA). The ground state energy and other characteristics for the infinite chain were also compared with those obtained earlier for a nanoscopic chain within the Exact Diagonalization combined with an Ab Initio adjustment of the single-particle wave functions in the correlated state (EDABI method). For the sake of completeness, we briefly characterize also each of the solutions. Our approach completes the Lieb-Wu solution, as it provides the system electronic properties evolution as a function of physically controllable parameter - the interatomic distance.
The optimized single-particle wave functions contained in the parameters of the Hubbard model (t and U ) were determined for an infinite atomic chain.In effect, the electronic properties of the chain as a function of interatomic distance R were obtained and compared for the Lieb-Wu exact solution, the Gutzwiller-wave-function approximation, and the Gutzwiller--ansatz case.The ground state energy and other characteristics for the infinite chain were also compared with those obtained earlier for a nanoscopic chain within the exact diagonalization combined with an ab initio adjustment of the single-particle wave functions in the correlated state (exact diagonalization combined with an ab initio method).For the sake of completeness, we briefly characterize also each of the solutions.Our approach completes the Lieb-Wu solution, as it provides the system electronic properties evolution as a function of physically controlable parameter -the interatomic distance.