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arXiv:1211.2408

Spin coherent states with monopole harmonics on the Riemann sphere for the Kravchuk oscillator

Zouhaı̈r Mouayn

Published Nov 11, 2012• 0 citations
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Abstract

We consider a class of generalized spin coherent states by choosing the labeling coefficients to be monopole harmonics.The latters are L2 eigenstates of the mth spherical Landau level on the Riemann sphere with m in Z+. We verify that the Klauder minimum properties for these states to be considered as coherent states are satisfied. We particularize them for the case of the Kravchuk oscillator and we obtain explicite expression for their wave functions.The associated coherent states transforms provide us with a Bargmann-type representation for the states of the oscillator Hilbert space. For the lowest level m = 0 indexing monopole harmonics, we identify the obtained coherent states to be those of Klauder-Perelomov type which were constructed in Ref.J. Math. Phys. 48, 112106 (2007)

DOI
10.48550/arxiv.1211.2408

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We consider a class of generalized spin coherent states by choosing the labeling coefficients to be monopole harmonics.The latters are L2 eigenstates of the mth spherical Landau level on the Riemann sphere with m in Z+. We verify that the Klauder minimum properties for these states to be considered as coherent states are satisfied. We particularize them for the case of the Kravchuk oscillator and we obtain explicite expression for their wave functions.The associated coherent states transforms provide us with a Bargmann-type representation for the states of the oscillator Hilbert space. For the lowest level m = 0 indexing monopole harmonics, we identify the obtained coherent states to be those of Klauder-Perelomov type which were constructed in Ref.J. Math. Phys. 48, 112106 (2007)
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While dealing with the J -matrix method for the harmonic oscillator to write down its tridiagonal matrix representation in an orthonormal basis of L 2 ( R ) , we rederive a set of generalized coherent states (GCS) of Perelomov type labelled by points z of the complex plane C and depending on an integer number m ∈ Z + . The number states expansion of these GCS gives rise to coefficients that are complex Hermite polynomials and whose linear superpositions provide eigenfunctions for the two-dimensional magnetic Laplacian, associated with the m th Landau level. We further generalize this construction to the Morse oscillator by introducing a new set of Glauber-type GCS.
Abstract. We describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d-dimensional sphere S d. The coherent states are labeled by points in the associated phase space T ∗ (S d). These coherent states are not of Perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators. We describe as well the Segal–Bargmann representation for the system, the associated unitary Segal–Bargmann transform, and a natural inversion formula. Although many of these results are in principle special cases of the results of B. Hall and M. Stenzel, we give here a substantially different description based on ideas of T. Thiemann and of K. Kowalski and J. Rembieliński. All of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space. We focus on the sphere case in order to carry out the calculations in a self-contained and explicit way.
We describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d-dimensional sphere Sd. The coherent states are labeled by points in the associated phase space T*(Sd). These coherent states are not of Perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators. We describe as well the Segal–Bargmann representation for the system, the associated unitary Segal–Bargmann transform, and a natural inversion formula. Although many of these results are in principle special cases of the results of Hall and Stenzel, we give here a substantially different description based on ideas of Thiemann and of Kowalski and Rembieliński. All of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space. We focus on the sphere case in order to carry out the calculations in a self-contained and explicit way.
We consider a particle moving on a 2-sphere in the presence of a constant magnetic field. Building on our earlier work in the nonmagnetic case we construct coherent states for this system. The coherent states are labeled by points in the associated phase space, the (co)tangent bundle of S 2 . They are constructed as eigenvectors for certain annihilation operators and expressed in terms of a certain heat kernel. These coherent states are not of Perelomov type but rather are constructed according to the ‘complexifier’ approach of Thiemann. We describe the Segal–Bargmann representation associated with the coherent states which is equivalent to a resolution of the identity. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.