Statistical approach to quantum mechanics I: General nonrelativistic theory

TL;DR

This paper establishes that any statistical description of nonrelativistic classical systems naturally leads to the Schrödinger equation, deriving the canonical quantization rule and discussing wave-particle compatibility.

Contribution

It introduces a general statistical framework for nonrelativistic quantum mechanics, deriving the Schrödinger equation and quantization rules from classical principles and probability conservation.

Findings

Derives the Schrödinger equation from classical statistical assumptions.

Establishes the canonical quantization rule based on the classical Hamiltonian.

Provides examples illustrating wave-particle duality and quantum phenomena.

Abstract

In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields the multi-coordinate Schr\"odinger equation, with its usual boundary conditions, as an essential statistical equation for the system. We derive the general "canonical quantization" rule, that the Hamiltonian operator must be the classical Hamiltonian in the $N$-dimensional metric configuration space defined by the classical kinetic energy of the system, with the classical conjugate momentum $N$-vector replaced by $-i\hbar$ times the vector gradient operator in that space. We obtain these results by using conservation of probability, general tensor calculus, the Madelung transform, the Ehrenfest theorem and/or the Hamilton-Jacobi equation, and comparison with results for the charged harmonic oscillator in stochastic electrodynamics. We also provide two illustrative examples and a discussion of how coordinate trajectories could be compatible with wave properties such as interference, diffraction, and tunneling.