We are concerned with an information-theoretic measure of uncertainty for quantum systems. Precisely, the Wehrl entropy of the phase-space probability Q (m) ρ = z, m|ρ|z, m which is known as Husimi function, whereρ is a density operator and |z, m are coherent states attached to an Euclidean mth Landau level. We obtain the Husimi function Q (m) β of the thermal density operator ρ β of the harmonic oscillator, which leads by duality, to the Laguerre probability distribution of the mixed light. We discuss some basic properties of Q (m) β such as its characteristic function and its limiting logarithmic moment generating function from which we derive the rate function of the sequence of probability distributions Q (m)β , m = 0, 1, 2, .... For m ≥ 1, we establish an exact expression for the Wehrl entropy of the density operatorρ β and we discuss the behavior of this entropy with respect to the temperature parameter T = 1/β.
We consider a one-parameter family of nonlinear coherent states by replacing the factorial in coefficients z n / √ n! of the canonical coherent states by a specific generalized factorial x γ n !, γ ≥ 0. These states are superposition of eigenstates of the Hamiltonian with a symmetric Pöschl-Teller potential depending on a parameter ν > 1. The associated Bargmann-type transform is defined for γ = ν. Some results on the infinite square well potential are also derived. For some different values of γ, we discuss two sets of orthogonal polynomials that are naturally attached to these coherent states.
We construct a new class of coherent states labeled by points z of the complex plane and depending on three numbers (γ, ν) and ε > 0 by replacing the coefficients z n / √ n! of the canonical coherent states by Laguerre polynomials. These states are superpositions of eigenstates of the symmetric Pöschl-Teller oscillator and they solve the identity of the states Hilbert space at the limit ε → 0 + . Their wavefunctions are obtained in a closed form for a special case of parameters (γ, ν). We discuss their associated coherent states transform which leads to an integral representation of Hankel type for Laguerre functions.
Bessel functions form an important class of special functions and are applied almost everywhere in mathematical physics. They are also called cylindrical functions, or cylindrical harmonics. This chapter is devoted to the construction of the generalized coherent state (GCS) and the theory of Bessel wavelets. The GCS is built by replacing the coefficient zn/n!,z∈C of the canonical CS by the cylindrical Bessel functions. Then, the Paley-Wiener space PW1 is discussed in the framework of a set of GCS related to the cylindrical Bessel functions and to the Legendre oscillator. We prove that the kernel of the finite Fourier transform (FFT) of L2-functions supported on −11 form a set of GCS. Otherwise, the wavelet transform is the special case of CS associated respectively with the Weyl-Heisenberg group (which gives the canonical CS) and with the affine group on the line. We recall the wavelet theory on R. As an application, we discuss the continuous Bessel wavelet. Thus, coherent state transformation (CST) and continuous Bessel wavelet transformation (CBWT) are defined. This chapter is mainly devoted to the application of the Bessel function.
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