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American Journal of Physics

A historical derivation of Heisenberg’s uncertainty relation is flawed

John H. Marburger

Published May 12, 2008• 2 citations
Publisher

Abstract

Kennard’s 1927 demonstration of the uncertainty relation, cited by Heisenberg as the earliest derivation using the formalism of quantum theory, invokes a trial function that severely limits the class of wave functions for which the uncertainty relation is shown to be valid.
DOI
10.1119/1.2870628

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Heisenberg's uncertainty principle has been understood to set a limitation on measurements; however, the long standing mathematical formulation established by Heisenberg, Kennard, and Robertson does not allow such an interpretation. Recently, a new relation was found to give a universally valid relation between noise and disturbance in general quantum measurements, and it has become clear that the new relation plays a role of the first principle to derive various quantum limits on measurement and information processing in a unified treatment. This paper examines the above development on the noise-disturbance uncertainty principle in the model independent approach based on measurement operator formalism, which is widely accepted to describe a class of generalized measurements in the field of quantum information. We obtain explicit formulas for noise and disturbance of measurements given by measurement operators, and show that projective measurements do not satisfy the Heisenberg-type noise-disturbance relation that is typical in the gamma-ray microscope thought experiments. We also show that the disturbance on a Pauli operator of a projective measurement of another Pauli operator constantly equals to the square root of 2, and examine how this measurement violates the Heisenberg-type relation but satisfies the new noise-disturbance relation.