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arXiv:0805.1331 · Progress of Theoretical Physics

Quantum States Allowing Minimum Uncertainty Product of and Lz

Tiago Pereira, Domingos H. U. Marchetti

Published Nov 1, 2009• 2 citations
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Abstract

We provide necessary and sufficient conditions for states to have an arbitrarily small uncertainty product of the azimuthal angle φ and its canonical moment Lz. We illustrate our results with analytical examples.
DOI
10.1143/ptp.122.1137

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The description of a quantum system in terms of angle variables may violate Heisenberg uncertainty principle. The familiar case is the azimutal angle ϕ\phiϕ and its canonical moment LzL_zLz​. Although this problem was foreseen almost a century ago, up to the present days there are no criteria to precisely characterize the violation. In this paper, we present a theorem which provides necessary and sufficient conditions for the violation of the Heisenberg uncertainty principle. We illustrate our results with analytical examples.
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The Heisenberg uncertainty relation, concerning the product of the standard deviations of position and momentum, is studied for a quantum oscillator characterized by deformed boson algebra. Special forms of the deformed number operator are determined such that the ground state and any deformed coherent state of the oscillator under study exhibit minimum uncertainty, as the canonical boson algebra provides, but the product of the corresponding standard deviations is less than the canonical lower bound, ℏ/2. Above all, each standard deviation and, consequently, the uncertainty become arbitrarily small for the ground state, other states of the Fock basis, and deformed coherent states with arbitrarily small magnitude of the label, if the parameters defining the special deformed number operator are properly chosen.