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Chemical Physics Letters

Schrödinger equation with Coulomb potential admits no exact solutions

Ilia Toli, Shengli Zou

Published Jan 1, 2019• 8 citations
Publisher

Abstract

We formally prove that Schrödinger equation with Coulomb potential has no nontrivial exact solutions for three or more unrestrained particles. New potentials are needed to solve this issue. This is the subject of an ongoing project. The proof technique is to logically embed three loops of mathematical induction into each other. Our work is an expansion of a work by Bartlett. It follows that the solutions proposed by Fock are not exact, although accurate to 40 digits.

DOI
10.1016/j.cpletx.2019.100021

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We formally prove that Schrödinger equation with Coulomb potential has no nontrivial exact solutions for three or more unrestrained particles. New potentials are needed to solve this issue. This is the subject of an ongoing project. The proof technique is to logically embed three loops of mathematical induction into each other. Our work is an expansion of a work by Bartlett. It follows that the solutions proposed by Fock are not exact, although accurate to 40 digits.
We consider the quantum-mechanical problem of a particle bound to a configuration of spherical potentials, each of finite range. If the Schr\"odinger equation can be solved for each potential by itself, then it is shown how to solve it for the configuration, provided the potentials do not overlap. The energy levels are the zeros of a determinant of formally infinite order, but in practice this is always well approximated by a finite determinant and often by one of small order. As illustrative examples we consider some states of a particle bound to three square wells with the configuration of an equilateral triangle, and to two truncated Coulomb potentials. The possibility of extending an approximate version of the method to overlapping potentials is pointed out.
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As the first stage in a project aimed at finding exact bound-state solutions for the few-particle Schrodinger equation, some restricted two- and threebody systems are treated by Rayleigh-Schrddinger perturbation theory. Formal solutions for the perturbation equations are obtained by using standard techniques for solving differential equations. Particular attention is given to justifying the various boundary conditions, since the same conditions apply to formal solutions of the eigenvalue equations. It is shown that boundary conditions designed to preserve the Hermitian character of the Hamiltonian are sufficient to determine all parameters in the formal solutions. The procedure is demonstrated by application to a hydrogen atom perturbed by stationary point charges. Results are given for the ground and some excited states.
A new approach to find exact solutions to one--dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions having vanishing and non--vanishing Bohm potentials. For most of the potentials, no solutions to the Schrödinger equation produce a vanishing Bohm potential. A (large but) restricted family of potentials allows the existence of particular solutions for which the Bohm potential vanishes. This family of potentials is determined, and several examples are presented. It is shown that some quantum, such as accelerated Airy wavefunctions, are due to the presence of non--vanishing Bohm potentials. New examples of this kind are found and discussed.
Introduction: an alternative solution to the Schrodinger-Langevin equation is presented, where the temporal dependence is explained, assuming a Coulomb potential. Finally, the trajectory equations are found. Objective: in this paper we contribute by presenting a detailed and simple solution of the Schrödinger-Langevin equation for a Coulomb potential. Materials and Methods: using an appropriate ansatz, we solve the Schrödinger-Langevin equation, finding the expected values of position and moment. Results: a simple method was presented to find the expected position and moment values in the Schrödinger-Langevin equation, the ansatz used to find these solutions allows the model to be generalized in a certain way to electric potentials and harmonic oscillators. Conclusions: the model used to solve the Schrödinger-Langevin equation, allowed to find the expected values of position and moment of a particle in a Coulomb potential, the temporal dependence of such solutions is made explicit, which allows finding the path equations of the particles.