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Theoretical and Mathematical Physics

Relationship between a quasipotential equation and a Schrödinger equation

Mikhail Aleksandrovich Braun

Published Sep 1, 1987• 9 citations
Publisher
DOI
10.1007/bf01018302

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In this paper I develop a relativistic Hamiltonian and Schr\"odinger equation for three particles using a generalization of the assumptions that give rise to Todorov's relativistic quasipotential equation for two particles. This formalism is treated by a method that leads to a generalization of the separable Lagrange equilateral-triangle solution from gravitational nonrelativistic to relativistic nongravitational bound-state problems in quantum as well as classical mechanics. As an example I derive an analytic solution for the spectrum of a three-body system bound by scalar Coulomb forces.
According to the energy relationship of the ground state solution of a hydrogen atom using the Schrödinger equation, the kinetic energy of an electron at twice the Bohr radius becomes zero. The electron has zero kinetic energy, which means it is dead, indicating that the energy relationship used in the Schrödinger equation is incorrect. Then, the reasons for both the chaotic energy relationship of the Schrödinger equation and the half frequency problem encountered by Bohr are analyzed. Bohr only considered Coulomb force and established an equation with centrifugal force, but he did not consider Lorentz force. So, the frequency he obtained must be multiplied by 1/2 to match the experimental value. This article also analyzes other issues related to the Schrödinger equation.