Why Probability appears in Quantum Mechanics

TL;DR

This paper presents a mathematical model demonstrating that the Copenhagen interpretation of quantum mechanics naturally arises from the unitary evolution of quantum systems and the probabilistic nature of large-scale measurements.

Contribution

It shows that probability in quantum mechanics can be derived from deterministic unitary evolution, providing a new perspective on the interpretation.

Findings

Copenhagen interpretation follows from unitary evolution and probabilistic measurement.

Probability enters quantum theory through large number of deterministic equations.

The model bridges deterministic quantum dynamics with probabilistic measurement outcomes.

Abstract

Early in the development of quantum theory Bohr introduced what came to be called the Copenhagen interpretation. Specifically, the square of the absolute value of the wave function was to be used as a probability density. There followed lengthy arguments about this ranging from alternative universes to Schr\"odinger's cat. Einstein famously remarked "I am convinced that He (God) does not play dice." The purpose of this paper is to present a mathematical model of the measuring process that shows that the Copenhagen interpretation can actually follow from the fact that the time development of quantum systems is governed by the usual one parameter group of unitary transformations exp(iHt) and that probability enters into the theory in the way it usually does in physics, namely, by having a large number of deterministic equations that can only be handled probabilistically