Foundations of quantum mechanics: non-relativistic theory

The objective of this paper is to axiomatically derive quantum mechanics from three basic axioms. In this paper, the Schrödinger equation for a characteristic function is first obtained and from it the Schrödinger equation for the probability amplitudes is also derived. The momentum and position operators acting upon the characteristic function are defined and it is then demonstrated that they do commute, while those acting upon the probability amplitudes obey the usual commutation relation. We also show that, for dispersion free ensembles, the Schrödinger equation for the characteristic function is equivalent to Newton's equations, thus providing us with a correspondence between both theories. As an application of the method, we show how it can be used to make quantization in generalized coordinates.