The Quantum Formalism Revisited
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.
Keywords
Cite
@article{arxiv.2506.16480,
title = {The Quantum Formalism Revisited},
author = {Hajo Leschke},
journal= {arXiv preprint arXiv:2506.16480},
year = {2026}
}
Comments
Differences from version v4: typos corrected, minor stylistic improvements, and three further references. To appear as a chapter of the book "Quantum Mechanics: A Century Later", ed. by Tuck C. Choy, World Scientific, Singapore 2026