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arXiv:quant-ph/0202070 · Journal of Physics A Mathematical and General

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

K. Kowalski, Jakub Rembieliński

Published Feb 4, 2002• 48 citations
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Abstract

The uncertainty relations for the position and momentum of a quantum particle on a circle are identified as minimized by the corresponding coherent states. The squeezed states in the case of the circular motion are introduced and discussed in the context of the uncertainty relations.
DOI
10.1088/0305-4470/35/6/306

Related papers

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 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.

arXiv:quant-ph/0209032•Journal of Physics A Mathematical and General•Feb 12, 2003•8 citations

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.

•Sep 4, 2002•7 citations

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

K. Kowalski, J Rembieli ski

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.

arXiv:quant-ph/0305069•Journal of Physics A Mathematical and General•May 8, 2003•12 citations

Schroedinger uncertainty relation and its minimization states

D. A. Trifonov

An introductory survey on the Schroedinger uncertainty relation and its minimization states is presented with minimal number of formulas and some historical points. The case of the two canonical observables, position and momentum, is discussed in greater detail: basic properties of the two subsets of minimization states (canonical squeezed and coherent states) are reviewed and compared. The case of two non-canonical observables is breafly outlined. Stanfard SU(1,1) and SU(2) group-related coherent states can be defined as states that minimize Schroedinger inequality for the three pairs of generators simultaneously. The symmetry of the Heisenberg and Schroedinger relations is also discussed, and two natural generalizations to the cases of several observables and several states are noted.

arXiv:physics/0105035•May 12, 2001•9 citations

On a subject of diverse improvisations: The uncertainty relations on a circle

Spiridon Dumitru

The disputed question of uncertainty relations (UR) on a circle is regarded as a particular element of a more general problem which refers to the quantum description of angular observables $L_z$ and $ϕ$. The improvised $L_z-ϕ$ UR are found to be affected by unsourmontable shortcomings. Also in contradiction with a largely accepted belief it is proved that the usual procedures of quantum mechanics are accurately applicable for the $L_z-ϕ$ pair. The applicability regards both the known circular motions and the less known non-circular rotational motions. The presented facts contribute as arguments to the following indubitable conclusions: (i) the traditional interpretation of UR must be denied as an incorrect doctrine, (ii) for a natural physical consideration of the the $L_z-ϕ$ pair the results from the usual quantum mechanics are sufficient while the improvised $L_z-ϕ$ UR must be rejected as senseless formulas and (iii) the descriptions of quantum measurements have to be done in a framework which is distinct and additional in respect with the usual quantum mechanics.

arXiv:quant-ph/0206009•Jun 3, 2002•4 citations

Coherent states on the circle

José A González, M. A. del Olmo

A careful study of the physical properties of a family of coherent states on the circle, introduced some years ago by de Bièvre and González (in 1992 Semiclassical behaviour of the Weyl correspondence on the circle Group Theoretical Methods in Physics vol I (Madrid: Ciemat)), is carried out. They were obtained from the Weyl-Heisenberg coherent states in by means of the Weil-Brezin-Zak transformation, they are labelled by the points of the cylinder , and they provide a realization of by entire functions (similar to the well known Fock-Bargmann construction). In particular, we compute the expectation values of the position and momentum operators on the circle and we discuss the Heisenberg uncertainty relation.

arXiv:quant-ph/9809020•Journal of Physics A Mathematical and General•Nov 6, 1998•73 citations
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.
An introductory survey on the Schroedinger uncertainty relation and its minimization states is presented with minimal number of formulas and some historical points. The case of the two canonical observables, position and momentum, is discussed in greater detail: basic properties of the two subsets of minimization states (canonical squeezed and coherent states) are reviewed and compared. The case of two non-canonical observables is breafly outlined. Stanfard SU(1,1) and SU(2) group-related coherent states can be defined as states that minimize Schroedinger inequality for the three pairs of generators simultaneously. The symmetry of the Heisenberg and Schroedinger relations is also discussed, and two natural generalizations to the cases of several observables and several states are noted.
The disputed question of uncertainty relations (UR) on a circle is regarded as a particular element of a more general problem which refers to the quantum description of angular observables LzL_zLz​ and ϕϕϕ. The improvised Lz−ϕL_z-ϕLz​−ϕ UR are found to be affected by unsourmontable shortcomings. Also in contradiction with a largely accepted belief it is proved that the usual procedures of quantum mechanics are accurately applicable for the Lz−ϕL_z-ϕLz​−ϕ pair. The applicability regards both the known circular motions and the less known non-circular rotational motions. The presented facts contribute as arguments to the following indubitable conclusions: (i) the traditional interpretation of UR must be denied as an incorrect doctrine, (ii) for a natural physical consideration of the the Lz−ϕL_z-ϕLz​−ϕ pair the results from the usual quantum mechanics are sufficient while the improvised Lz−ϕL_z-ϕLz​−ϕ UR must be rejected as senseless formulas and (iii) the descriptions of quantum measurements have to be done in a framework which is distinct and additional in respect with the usual quantum mechanics.
A careful study of the physical properties of a family of coherent states on the circle, introduced some years ago by de Bièvre and González (in 1992 Semiclassical behaviour of the Weyl correspondence on the circle Group Theoretical Methods in Physics vol I (Madrid: Ciemat)), is carried out. They were obtained from the Weyl-Heisenberg coherent states in by means of the Weil-Brezin-Zak transformation, they are labelled by the points of the cylinder , and they provide a realization of by entire functions (similar to the well known Fock-Bargmann construction). In particular, we compute the expectation values of the position and momentum operators on the circle and we discuss the Heisenberg uncertainty relation.