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American Journal of Physics

Uncertainty Principle and Normal Operators—Some Comments

Wesley E. Brittin

Published Oct 1, 1966• 2 citations
Publisher

Abstract

It is shown that an appropriate formulation of the uncertainty principle for normal operators leads to the same mathematical expression as that for Hermitian operators. The treatment is valid for mixed states as well as for pure states.
DOI
10.1119/1.1972318

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The uncertainty relations (URs) of two arbitrary Hermitian and non-Hermitian incompatible operators represented by the product of variances have been confirmed theoretically and experimentally in various physical systems. However, the lower bound of the product uncertainty inequality can be null even for two noncommuting operators, i.e., a trivial case. Therefore, for two incompatible operators over the measured system state, the associated URs regarding the sum of variances are valid in a state-dependent manner, and the lower bound is guaranteed to be nontrivial. Although the sum URs formulated for Hermitian and unitary operators have been affirmed, the general forms for arbitrary non-Hermitian operators have not yet been investigated. This study presents the sum URs for non-Hermitian operators acting on system states using an appropriate Hilbert-space metric. The compatible forms of our sum inequalities with the conventional quantum mechanics are also provided via the GGG-metric formalism. Concrete examples illustrate the validity of the proposed sum URs in both PT\mathcal{PT}PT-symmetric and PT\mathcal{PT}PT-broken phases. The developed methods and results can help give an in-depth understanding of the usefulness of GGG-metric formalism in non-Hermitian quantum mechanics and the sum URs of incompatible operators within.