The Uncertainty Principle as an Eigenwert Problem

A. Gamba

GREGORY A. BREIT

Using Weyl's geometrical interpretation of quantum mechanics the fundamental relations of the transformation theory can be treated without making use of Dirac's $\ensuremath{\delta}$ function. The principle of uncertainty is particularly clear in this system. It is shown that in the discrete representation for a definite axis of the coordinate $q$ all Eigenwerte of the conjugate momentum $p$ are equally probable. It is also shown how in the limit of very fine subdivision the results of Heisenberg and Bohr are obtained. The whole treatment receives an easy interpretation by applying it to the problem of the diffraction of waves by a grating. Weyl's method of discussing quantities with continuous ranges of values is found to be equivalent to investigating diffraction phenomena in the limiting case of an infinitesimal grating space. The quantum condition $i(PQ\ensuremath{-}QP)=1$ is shown to be correct in a purely operational sense in spite of the fact that as a matrix equation it is untrue.

Jong, Chol, Shin-Hyok Jon, Son-Il Jo +1 more

The uncertainty principle is the most important for the foundations of quantum mechanics but it still remains failed to reach a consensus of its interpretation, which gives rise to debates upon its physical nature. In this work, we address the problem of its foundation from a different aspect to present an alternative formulation for proving the uncertainty relations in a general way in terms of commutation relations and action function. The relationship between the de Broglie relation and the uncertainty principle is studied from a new angle. As a result, it is demonstrated that the de Broglie relation is the foundation of the uncertainty principle. Starting with the origin of the problem, we show that the de Broglie relation provides the form of the wave function and the determined form of the wave function in turn leads to the conception of operators for quantum mechanics, and thus it is possible to provide with the help of the operators and wave function a generalized proof of the uncertainty principle as the law governing ensemble of states. As a decisive solution to the problem, the interpretation of the uncertainty principle in terms of the action function is offered that gives one self-consistent explanation in agreement with the known physical phenomena. Eventually, we show the necessity and possibility of reassessing and improving the foundation and interpretation of the uncertainty principle as a leading principle of quantum mechanics.

Steven Matthysse

Bin Liang

This article discusses the uncertainty problem in the classical mechanics, believes the uncertainty principle to be a fundamental principle in the classcal mechanics, as in the quantum mechanics.

Steeve Zozor, M. Portesi, Christophe Vignat