The Schrödinger Equation with Coulomb Potential Admits no Exact Solutions

ABSTRACT 

We prove that theSchrödinger equationwith the electrostatic potential energy expressed by theCoulomb potentialdoes not admit exact solutions for three or more bodies. It follows that theexact solutions proposed by Fock [1–3] are flawed. The Coulomb potential is the problem. Basedon the classical (non-quantum)principle of superposition, the Coulomb potential of a system ofmany particles is assumed to be the sum of all the pairwise Coulomb potentials. We prove that thisis not accurate. The Coulomb potential being a hyperbolic (not linear) function, the superpositionprinciple does not apply.The Schrödinger equation as studied in this PhD dissertation is a linearpartialdifferential equationwith variable coefficients. The only exception is the Schrödinger equation for the hydrogen atom,which is a linearordinarydifferential equation with variable coefficients. No account is kept of thespin or the effects of the relativity.New electrostatic potentials are proposed for which the exact solutions of the Schrödinger equationexist. These new potentials obviate the need for the three-body force [4] interpretations of theelectrostatic potential.Novel methods for finding the exact solutions of the differential equations are proposed. Novel prooftechniques are proposed for the nonexistence of the exact solutions of the differential equations,be they ordinary or partial, with constant or variable coefficients. Few novel applications of theestablished approximate methods of the quantum chemistry are reported. They are simple from theviewpoint of the quantum chemistry, but have some important aerospace engineering applications.

 

 https://stars.library.ucf.edu/cgi/viewcontent.cgi?article=7585&context=etd