A Wave Function that Violates the Heisenberg Uncertainty Principle

Keiichi Takamizawa

Published May 29, 2026• 0 citations

Abstract

A one-dimensional wave function that is defined in a bounded close interval violates the Heisenberg uncertainty principle. It was analyzed in detail. The author examined the properties of Hermitian operators and reported that an equation is the origin of the violation of the uncertainty relationship. It was also found that the boundary conditions play a crucial role in the violation of the uncertainty principle, i.e., the periodic boundary condition and the Dirichlet condition, which make a wave function zero at boundaries, yield different results. A novel elementary proof of Robertson’s uncertainty relationship, which does not use the Cauchy-Schwarz inequality, was provided in the present paper. The paper notes that if the commutation relationship between two Hermitian operators is given, there might be a case in which the fundamental properties may not be necessarily satisfied. Therefore, even if a noncommutative commutation relationship is given, the uncertainty relationship might not be valid.

DOI: 10.5281/zenodo.20447952

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