The Quantum Formalism Revisited

Abstract: For the simple system of a point-like particle in the real line I compile and contrast, starting with a concise table, the structural elements of quantum mechanics with those of classical (statistical) mechanics. Despite many amazing similarities, there are the well-known fundamental differences. The basic reason for all of them is the algebraic non-commutativity in the quantal structure. It was discovered by Werner Heisenberg (1901-1976) in June 1925 on the small island of Helgoland in the North Sea, as a consequence of his success at understanding atomic spectral data within a matrix scheme which is consistent with energy conservation. I discuss the differences and exemplify quantifications of them 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.