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arXiv:2003.04306

Understanding correlation between two bound state electrons using a simple model

Verma, Shivani, Aniruddha Chakraborty

Published Mar 8, 2020• 0 citations
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Abstract

Understanding electron correlation requires solving inseparable Schrodinger equation. In general, inseparable Schrödinger equations cannot be solved analytically. So their solutions are obtained numerically. In this paper we investigate electron correlation problem using the Dirac delta function repulsion between two electrons, where each electron is bound in their respective Dirac delta potential well.

DOI
10.48550/arxiv.2003.04306

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Understanding electron correlation requires solving inseparable Schrodinger equation. In general, inseparable Schrödinger equations cannot be solved analytically. So their solutions are obtained numerically. In this paper we investigate electron correlation problem using the Dirac delta function repulsion between two electrons, where each electron is bound in their respective Dirac delta potential well.
Abstract Inseparable Schrödinger equations usually cannot be solved analytically. Most often their solutions are obtained numerically. It would be helpful to have an inseparable Schrödinger equation whose solution can be completely calculated. Perhaps more can be learned about appropriate wave function solutions of complex problems if the solutions of an inseparable Schrödinger equation, solved in terms of simple functions, are available. Such a study is possible using the Dirac delta function as the potential energy for two particles moving on a line. The Dirac delta function problem leads naturally to a consideration of the wave functions' symmetry properties. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
In a previous note (1), it was noted one could write a solution of the bound state Schrodinger equation as W(x)=Wo(x)Dn(x) where W(x) is the wavefunction and Wo(x), the ground state wavefunction. Inserting into the Schrodinger equation one may obtain the Schrodinger equation for Wo(x) and an equation coupling Wo(x) and Dn(x). In this note, we suggest such an approach may be applied to the one dimensional bound Dirac equations. This allows one to obtain decoupled equations for u1(x) and v1(x) where u(x)=uo(x)u1(x) and v(x)=v1(x)vo(x) are the full solutions. Thus, if one is able to obtain a solution for uo(x) and vo(x) by inspection, the decoupled equations allow one to find the general solution. We apply this approach to the case of a scalar potential g|x|. The Dirac problem for this potential has been solved in the literature (2) with decoupling occurring through the taking of spatial derivatives. Here we wish to suggest a general method.
The problem of bound states in delta potentials is revisited by means of Fourier transform approach. The problem in a simple delta potential sums up to solve an algebraic equation of degree one for the Fourier transform of the eigenfunction and the problem for more than one delta function also reveals itself to be a simple matter. Quite differently from direct methods, no knowledge about the jump discontinuity of the first derivative of the eigenfunction is required to determine the solution of the problem.