Uncertainty principle and uncertainty relations

J. B M Uffink, J. Hilgevoord

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It is generally believed that the uncertainty relation Δq Δp≥1/2h{combining short stroke overlay}, where Δq and Δp are standard deviations, is the precise mathematical expression of the uncertainty principle for position and momentum in quantum mechanics. We show that actually it is not possible to derive from this relation two central claims of the uncertainty principle, namely, the impossibility of an arbitrarily sharp specification of both position and momentum (as in the single-slit diffraction experiment), and the impossibility of the determination of the path of a particle in an interference experiment (such as the double-slit experiment). The failure of the uncertainty relation to produce these results is not a question of the interpretation of the formalism; it is a mathematical fact which follows from general considerations about the widths of wave functions. To express the uncertainty principle, one must distinguish two aspects of the spread of a wave function: its extent and its fine structure. We define the overall width WΨ and the mean peak width wψ of a general wave function ψ and show that the product WΨwφ is bounded from below if φ is the Fourier transform of ψ. It is shown that this relation expresses the uncertainty principle as it is used in the single- and double-slit experiments.

Original languageEnglish (US)
Pages (from-to)925-944
Number of pages20
JournalFoundations of Physics
Issue number9
StatePublished - Sep 1985

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