We prove that the Schrödinger equation with the electrostatic potential energy expressed by the Coulomb potential does not admit exact solutions for three or more bodies. It follows that the exact solutions proposed by Fock [1–3] are flawed. The Coulomb potential is the problem. Based on the classical (non-quantum) principle of superposition, the Coulomb potential of a system of many particles is assumed to be the sum of all the pairwise Coulomb potentials. We prove that this is not accurate. The Coulomb potential being a hyperbolic (not linear) function, the superposition principle does not apply. The Schrödinger equation as studied in this PhD dissertation is a linear partial differential equation with variable coefficients. The only exception is the Schrödinger equation for the hydrogen atom, which is a linear ordinary differential equation with variable coefficients. No account is kept of the spin or the effects of the relativity. New electrostatic potentials are proposed for which the exact solutions of the Schrödinger equation exist. These new potentials obviate the need for the three-body force  interpretations of the electrostatic potential. Novel methods for finding the exact solutions of the differential equations are proposed. Novel proof techniques 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 the established approximate methods of the quantum chemistry are reported. They are simple from the viewpoint of the quantum chemistry, but have some important aerospace engineering applications.
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Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Toli, Ilia, "The Schrödinger Equation with Coulomb Potential Admits no Exact Solutions" (2019). Electronic Theses and Dissertations. 6585.