The Quantum Formalism Revisited

TL;DR

This paper compares the structural elements of quantum and classical mechanics for a simple particle system, highlighting fundamental differences due to non-commutativity and illustrating them with various inequalities and bounds.

Contribution

It provides a concise compilation and comparison of quantum and classical mechanics structures, emphasizing the algebraic non-commutativity in quantum theory.

Findings

Quantum mechanics differs from classical mechanics due to non-commutativity.

Variance and entropic inequalities quantify quantum indeterminacy.

Bell inequalities highlight non-classical correlations.

Abstract

For the simple system of a point-like particle confined to a straight line, I compile, initially in a concise table, the structural elements of quantum mechanics and contrast them with those of classical (statistical) mechanics. Despite many similarities, there are the well-known fundamental differences, resulting from the algebraic non-commutativity in the quantal structure. The latter was discovered by Werner Heisenberg (1901-1976) in June 1925 on the small island of Helgoland in the North Sea, as a consequence of understanding atomic spectral data within a matrix scheme consistent with energy conservation. I discuss the differences and exemplify their quantifications by the variance and entropic indeterminacy inequalities, by (pseudo-)classical bounds on quantum canonical partition functions, and by the correlation inequalities of John Bell (1928-1990) and others.