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arXiv:quant-ph/0209032 · Journal of Physics A Mathematical and General

Comment on On the uncertainty relations and squeezed states for the quantum mechanics on a circle

D A Trifonov

Published Feb 12, 2003• 8 citations
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Abstract

It is shown by examples that the position uncertainty on a circle, proposed recently by Kowalski and Rembieliński (2002 J. Phys. A:Math. Gen. 35 1405 ) is not consistent with the state localization. We argue that the relevant uncertainties and uncertainty relations (URs) on a circle are those based on the Gram–Robertson matrix. Several of these generalized URs are displayed and related criteria for squeezed states are discussed.

DOI
10.1088/0305-4470/36/8/316

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Comment on "On the uncertainty relations and squeezed states for the quantum mechanics on a circle"

D. A. Trifonov

It is shown by examples that the position uncertainty on a circle, proposed recently by Kowalski and Rembieli\\'nski [J. Phys. A 35 (2002) 1405] is not consistent with the state localization. We argue that the relevant uncertainties and uncertainty relations (UR's) on a circle are that based on the Gram-Robertson matrix. Several of these generalized UR's are displayed and related criterions for squeezed states are discussed.

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Reply to the Comment on On the uncertainty relations and squeezed states for the quantum mechanics on a circle

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In the preceding Comment (quant-ph/0209032) Trifonov disputes our recently proposed uncertainty relations for a quantum particle on a circle. He states that (i) the quantity $Δ^2(\hatϕ)$ introduced by us representing the uncertainty of the angle is not a proper measure of the position uncertainty and therefore the proposed inequalities can hardly be qualified as a relevant uncertainty relations on a circle; and that (ii) the most suitable uncertainty relations on a circle are those based on the Gram-Robertson matrix. It is shown by examples that both points are erroneous.

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Errors were introduced at typesetting stage to equations (11) and (12) on page 2201. Please see the pdf for the corrected formulas.

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It is shown by examples that the position uncertainty on a circle, proposed recently by Kowalski and Rembieliński (2002 J. Phys. A:Math. Gen. 35 1405 ) is not consistent with the state localization. We argue that the relevant uncertainties and uncertainty relations (URs) on a circle are those based on the Gram–Robertson matrix. Several of these generalized URs are displayed and related criteria for squeezed states are discussed.
In the preceding Comment (quant-ph/0209032) Trifonov disputes our recently proposed uncertainty relations for a quantum particle on a circle. He states that (i) the quantity Δ2(ϕ^)Δ^2(\hatϕ)Δ2(ϕ^​) introduced by us representing the uncertainty of the angle is not a proper measure of the position uncertainty and therefore the proposed inequalities can hardly be qualified as a relevant uncertainty relations on a circle; and that (ii) the most suitable uncertainty relations on a circle are those based on the Gram-Robertson matrix. It is shown by examples that both points are erroneous.
We consider the uncertainty relation between position and momentum of a particle on $ S^1 $ (a circle). Since $ S^1 $ is compact, the uncertainty of position must be bounded. Consideration on the uncertainty of position demands delicate treatment. Recently Ohnuki and Kitakado have formulated quantum mechanics on $ S^D $ (a DDD-dimensional sphere). Armed with their formulation, we examine this subject. We also consider parity and find a phenomenon similar to the spontaneous symmetry breaking. We discuss problems which we encounter when we attempt to formulate quantum mechanics on a general manifold.