Wave Mechanics without Uncertainty

Antony J. Bourdillon

Philip L. Bowers

Louis de Broglie

André Michaud

Analysis of the initial stages of the logical process followed by Louis de Broglie in establishing the electron phase wave equation in his 1924 thesis, which triggered the development of Wave Mechanics when Erwin Schrödinger formalized this concept with his vectorial wave equation. This development was soon followed by Quantum Mechanics, when Schrödinger proved that the Matrix Mechanics independently developed by Werner Heisenberg was equivalent to Wave Mechanics, with both theories leaving room for some degree of uncertainty as to the physical localization of the moving electron. This is what led Heisenberg to also formalize the Uncertainty Principle to take this situation into account. This principle was soon regarded as a fundamental axiomatic principle that seemed to make further exploration of the subatomic level of magnitude appear impossible to most researchers. We will analyze in this article the reason why the phase-wave velocity established by de Broglie generated this uncertainty in the localization of the moving electron in light of the current state of knowledge on the behavior of the electron in motion, in view of establishing the relevance of maintaining the Uncertainty Principle in the study of the subatomic level of magnitude.

Iwo Biatynicki-Birula, Jerzy Mycielski

New uncertainty relations in quantum mechanics are derived. They express restrictions imposed by quantum theory on probability distributions of canonically conjugate variables in terms of corresponding information entropies. The Heisenberg uncertainty relation follows from those inequalities and so does the Gross-Nelson inequality. The purpose of this paper is to derive a new, stronger version of the Heisenberg uncertainty relation in wave mechanics. This new uncertainty relation has a simple interpretation in terms of information theory. It is also closely related to newly discovered logarithmic Sobolev inequalities. The new uncertainty relation has the form (for wave functions normalized to unity)

J.P. MATHIEU