Simple derivation of Schrodinger equation from Newtonian dynamics

Abstract

The Eherenfest theorem states that Schrodinger representation of quantum mechanics (wave mechanics) reproduces Newton laws of motion in terms of expectation values. Remarkably, the contrary is considered elusive and, indeed, many authors have tried to obtain wave mechanics starting from other alternative frameworks of classical mechanics (for instance, Hamilton-Jacobi theory). Despite this common opinion, we present here a simple method to make Newtonian dynamics develop naturally into Schrodinger representation. The proof is based on the assumption of matter waves and is laid out in three fundamental steps. First, the role of classical density functions is underlined in view of their use to define constants of the motion for massive particles. Thanks to this preparatory step, density functions generate wave functions whose spatial and time variables obey Newton laws of motion. The resulting wave equation is defined in dependence on a parameter that plays the identical role of the constant K introduced by Schrodinger in the original formulation of his theory. In the final step, the classical wave equation is treated under the hypothesis of conservative forces common to the Eherenfest theorem and, after some algebra, the Schrodinger equation emerges by means of the identification of the classical momentum with de Broglie momentum of matter waves.