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Coherent states for a quantum particle on a circle

K. Kowalski, Jakub Rembieliński, Loucas C. Papaloucas

Published Jan 1, 1996• 91 citations
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Abstract

Abstract. The coherent states for the quantum particle on the circle are introduced. The Bargmann representation within the actual treatment provides the representation of the algebra [ ˆ J, U] = U, where U is unitary, which is a direct consequence of the Heisenberg algebra [ˆϕ, ˆ J] = i, but it is more adequate for the study of the circlular motion. Key words: quantum mechanics, coherent states, quantum groups

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Coherent states for a quantum particle on a circle

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Coherent states on the circle

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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 [DG 92], is carried out. They were obtained from the Weyl-Heisenberg coherent states in $L^2(\\R)$ by means of the Weil-Brezin-Zak transformation, they are labeled by the points of the cylinder $S^1 \\times \\R$, and they provide a realization of $L^2(S^1)$ 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.

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The Bargmann representation for the quantum mechanics on a sphere

K. Kowalski, J. Rembieliński

The Bargmann representation is constructed corresponding to the coherent states for a particle on a sphere introduced in Kowalski and Rembieliński, J. Phys. A: 33, 6035 (2000). The connection is discussed between the introduced formalism and the standard approach based on the Hilbert space of square integrable functions on a sphere S2.

arXiv:quant-ph/0011070•Journal of Mathematical Physics•Sep 1, 2001•22 citations
Abstract. The coherent states for the quantum particle on the circle are introduced. The Bargmann representation within the actual treatment provides the representation of the algebra [ ˆ J, U] = U, where U is unitary, which is a direct consequence of the Heisenberg algebra [ˆϕ, ˆ J] = i, but it is more adequate for the study of the circlular motion. Key words: quantum mechanics, coherent states, quantum groups
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.
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 [DG 92], is carried out. They were obtained from the Weyl-Heisenberg coherent states in L2(R)L^2(\\R)L2(R) by means of the Weil-Brezin-Zak transformation, they are labeled by the points of the cylinder S1timesRS^1 \\times \\RS1timesR, and they provide a realization of L2(S1)L^2(S^1)L2(S1) 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.
It is shown that the representation matrices of the quantum mechanical group in the coherent state basis can be used as bases for the expansion of arbitrary square integrable functions on the group. Although this basis is nonorthogonal the expansion can be inverted in a manner similar to an orthogonal basis. Next a coherent state ‘‘wave function’’ is defined in analogy with the Schrödinger wave function and the above matrix elements are used to determine the action of the group in the space of the coherent state wave functions. The function space in this realization is Bargmann’s Hilbert space of analytic functions. It is shown that the mixed basis matrix element between the coordinate and coherent state bases is essentially the integral kernel of Bargmann. The transpose of this matrix element yields the kernel for the inversion of the transform. In this construction the Bargmann transform appears as the unitary transformation connecting the Schrödinger and coherent state wave functions. This considerably simplifies Bargmann’s original arguments and yields the integral transform as well as its inversion formula in a simple way.