The SU(1,1) coherent states for a relativistic model of the linear singular oscillator are considered. The corresponding partition function is evaluated. The path integral for the transition amplitude between SU(1,1) coherent states is given. Classical equations of the motion in the generalized curved phase space are obtained. It is shown that the use of quasiclassical Bohr-Sommerfeld quantization rule yields the exact expression for the energy spectrum.
A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is presented. A natural approach to building coherent states associated to deformed algebras is then deduced.
Classes of coherent states are presented by replacing the labeling parameter $z$ of Klauder-Perelomov type coherent states by confluent hypergeometric functions with specific parameters. Temporally stable coherent states for the isotonic oscillator Hamiltonian are presented and these states are identified as a particular case of the so-called Mittag-Leffler coherent states.
The aim of this paper is to study generalized q-analogs of the well-known q-deformed harmonic oscillators and to connect them with q-Hermite polynomials. We give a construction of the appropriate oscillator-like algebras and show that corresponding Hermite polynomials are generalization of the discrete q-Hermite I and the discrete q-Hermite II polynomials. We also construct generalized coherent states of Barut-Girardello type for oscillator-like systems connected with these polynomials.
A dynamical algebra ${\cal A}_q$, englobing many of the deformed harmonic oscillator algebras is introduced. One of its special cases is extensively developed. A general method for constructing coherent states related to any algebra of the type ${\cal A}_q$ is discussed. The construction following this method is carried out for the special case.
$C_λ$-extended oscillator algebras, generalizing the Calogero-Vasiliev algebra, where $C_λ$ is the cyclic group of order $λ$, have recently proved very useful in the context of supersymmetric quantum mechanics and some of its variants. Here we determine the spectrum generating algebra of the $C_λ$-extended oscillator. We then construct its coherent states, study their nonclassical properties, and compare the latter with those of standard $λ$-photon coherent states, which are obtained as a special case. Finally, we briefly review some other types of coherent states associated with the $C_λ$-ext…
We construct a new class of coherent states indexed by points z of the complex plane and depending on two positive parameters m and epsilon by replacing the coefficients of the canonical coherent states by polyanalytic functions. These states solve the identity of the states Hilbert space of the harmonic oscillator at the limit epsilon goes to zero and obey a thermal stability property. Their wavefunctions are obtained in a closed form and their associated Bargmann-type transform is also discussed.
Using the formalism of Maya diagrams and ladder operators, we describe the algebra of annihilating operators for the class of rational extensions of the harmonic oscillators. This allows us to construct the corresponding coherent state in the sense of Barut and Girardello. The resulting time-dependent function is an exact solution of the time-dependent Schrodinger equation and a joint eigenfunction of the algebra of annihilators.
We construct two-parameters family of nonlinear coherent states by replacing the factorial in coefficients $z^n/\sqrt{n!}$ of the canonical coherent states by a specific generalized factorial $x_n^{γ,σ}!$ where parameters $γ$ and $σ$ satisfy some conditions for which the normalization condition and the resolution of identity are verified. The obtained family is a generalization of the Barut-Girardello coherent states and those of the philophase states. In the particular case of parameters $γ$ and $σ$, we attache these states to the pseudo-harmonic oscillator depending on two parameters $α,β> 0…
In the frame of our approach we constructed the generalized oscillator connected with Krawtchouk polynomials (named Krawtchouk oscillator) and coherent states for this oscillator too. Ours results are compared with analogues ones obtained for another variant of Krawtchouk oscillator in the paper [N.M.Atakishiev and S.K.Suslov, Difference analogs of the harmonic oscillator, Teor. Mat. Fiz., 85, 64-73 (1990)] from other point of view. Our definition of coherent states is close to one given in [B.Roy and P.Roy, Phase properties of a new nonlinear coherent state, quant-ph/0002043] This investigati…
A set of generalized squeezed-coherent states for the finite u(2) oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the su(2) generators. These matrix elements are given in the (N+1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3x3 matrix multi-orthogonal polynomials Q_n(k) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomi…
We construct a class of generalized phase coherent states indexed by points of the unit circle and depending on three positive parameters "gamma","alpha" and "epsilon" by replacing the labelling coefficients of the canonical coherent states by circular Jacobi polynomials with parameter "gamma". The special case "gamma" = 0 corresponds to well known phase coherent phase states. The constructed states are superposition of eigenstates of a one-parameter pseudoharmonic oscillator depending on "alpha" and solve the identity of the state Hilbert space at the limit "epsilon"->0+. Closed form for thei…
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…
Exactly solvable $N$-dimensional model of the quantum isotropic singular oscillator in the relativistic configurational $\vec r_N$-space is proposed. It is shown that through the simple substitutions the finite-difference equation for the $N$-dimensional singular oscillator can be reduced to the similar finite-difference equation for the relativistic isotropic three-dimensional singular oscillator. We have found the radial wavefunctions and energy spectrum of the problem and constructed a dynamical symmetry algebra.
Wave packets for the Quantum Non-Linear Oscillator are considered in the Generalized Coherent State framerwork. To first order in the non-linearity parameter the Coherent State behaves very similarly to its classical counterpart. The position expectation value oscillates in a simple harmonic manner. The energy-momentum uncertainty relation is time independent as in a harmonic oscillator. Various features, (such as the Squeezed State nature), of the Coherent State have been discussed.
For a q-deformed harmonic oscillator, we find explicit coordinate representations of the creation and annihilation operators, eigenfunctions, and coherent states (the last being defined as eigenstates of the annihilation operator). We calculate the product of the coordinate momentum uncertainties in qoscillator eigenstates and in coherent states. For the oscillator, this product is minimum in the ground state and equals 1/2, as in the standard quantum mechanics. For coherent states, the $q$-deformation results in a violation of the standard uncertainty relation; the product of the coordinate- …
Starting from a faithful five-dimensional matrix representation of the group of two independent oscillators and applying the R-matrix method we generate some classes of deformed fermionic-bosonic quantum Hopf algebras. The corresponding Lie deformed superalgebras of type I--II, obtained by duality, are computed and a realization of generators of these deformed superalgebras are given in terms of the usual fermionic and bosonic creation and annihilation operators associated to the supersymmetric harmonic oscillator. Then, a generalized deformed annihilator is construted and their eigenstates ar…
Vector coherent states (VCS) viewed as a generalization of ordinary coherent states for higher rank tensor Hilbert spaces are investigated. We consider a systematic way of generating classes of VCS which are solvable (i.e., in the present context, normalizable states satisfying a resolution of the identity) on the Hilbert space of 2D and 3D harmonic oscillators. Thanks to the type of construction, these VCS are classified according to specific criteria. Furthermore, in many cases, the found classes of VCS are continuously deformable one onto another, still remaining solvable.
We have constructed coherent states for the higher derivative Pais-Uhlenbeck Oscillator. In the process we have suggested a novel way to construct coherent states for the oscillator having only negative energy levels. These coherent states have negative energies in general but their coordinate and momentum expectation values and dispersions behave in an identical manner as that of normal (positive energy) oscillator. The coherent states for the Pais-Uhlenbeck Oscillator have constant dispersions and a modified Heisenberg Uncertainty Relation. Moreover, under reasonable assumptions on parameter…
We construct a new class of coherent states labeled by points z of the complex plane and depending on three numbers (gamma, nu) and epsilon positive by replacing the coefficients of the canonical coherent states by Laguerre polynomials. These states are superpositions of eigenstates of the symmetric Poschl-Teller oscillator and they solve the identity of the states Hilbert space at the limit epsilon goes to 0. Their wavefunctions are obtained in a closed form for a special case of parameters (gamma, nu). We discuss their associated coherent states transform which leads to an integral represent…
The coherent states for twist-deformed oscillator model provided in article [1] are constructed. Besides, it is demonstrated that the energy spectrum of considered model is labeled by two quantum numbers - by so-called main and azimutal quantum numbers respectively.
The coherent state of a nonlinear oscillator having a nonlinear spectrum is constructed using Gazeau Klauder formalism. The weighting distribution and the Mandel parameter are studied. Details of the revival structure arising from different time scales underlying the quadratic energy spectrum are investigated by the phase analysis of the autocorrelation function.
We construct coherent state of the effective mass harmonic oscillator and examine some of its properties. In particular closed form expressions of coherent states for different choices of the mass function are obtained and it is shown that such states are not in general x-p uncertainty states. We also compute the associated Wigner functions.
In this paper, we propose a full characterization of a generalized $q-$deformed Tamm-Dancoff oscillator algebra and investigate its main mathematical and physical properties. Specifically, we study its various representations and find the condition satisfied by the deformed $q-$number to define the algebra structure function. Particular Fock spaces involving finite and infinite dimensions are examined. A deformed calculus is performed as well as a coordinate realization for this algebra. A relevant example is exhibited. Associated coherent states are constructed. Finally, some thermodynamics a…
We consider a one-parameter family of nonlinear coherent states by replacing the factorial in coefficients of the canonical coherent states by a specific generalized factorial depending on a parameter gamma. These states are superposition of eigenstates of the Hamiltonian with a symmetric Poschl-Teller potential depending on a parameter nu > 1. The associated Bargmann-type transform is defined for equal parameters. Some results on the infinite square well potential are also derived. For some different values of gamma, we discuss two sets of orthogonal polynomials that are naturally attached to…
We explore squeezed coherent states of a 3-dimensional generalized isotonic oscillator whose radial part is the newly introduced generalized isotonic oscillator whose bound state solutions have been shown to admit the recently discovered $X_1$-Laguerre polynomials. We construct a complete set of squeezed coherent states of this oscillator by exploring the squeezed coherent states of the radial part and combining the latter with the squeezed coherent states of the angular part. We also prove that the three mode squeezed coherent states resolve the identity operator. We evaluate Mandel's $Q$-par…
We build the coherent states for a family of solvable singular Schrödinger Hamiltonians obtained through supersymmetric quantum mechanics from the truncated oscillator. The main feature of such systems is the fact that their eigenfunctions are not completely connected by their natural ladder operators. We find a definition that behaves appropriately in the complete Hilbert space of the system, through linearised ladder operators. In doing so, we study basic properties of such states like continuity in the complex parameter, resolution of the identity, probability density, time evolution and po…
We discuss a model of a $q$-harmonic oscillator based on Rogers-Szegő functions. We combine these functions with a class of $q$-analogs of complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative integer parameter $m$. Our construction leads to a new $q$-deformation of the $m$-true-polyanalytic Bargmann transform whose range defines a generalization of the Arik-Coon space. We also give an explicit formula for the reproducing kernel of this space. The obtained results may be exploited to define a $q$-deformation of the Ginibre-$m$-type process on the comp…
Bound-state solutions of the singular harmonic oscillator and singular Coulomb potentials in arbitrary dimensions are generated in a simple way from the solutions of the one-dimensional generalized Morse potential. The nonsingular harmonic oscillator and nonsingular Coulomb potentials in arbitrary dimensions with their additional accidental degeneracies are obtained as particular cases. Added bonuses from these mappings are the straightforward determination of the critical attractive singular potential, the proper boundary condition on the radial eigenfunction at the origin and the inexistence…
An exact approach for the factorization of the relativistic linear singular oscillator is proposed. This model is expressed by the finite-difference Schrödinger-like equation. We have found finite-difference raising and lowering operators, which are with the Hamiltonian operator form the close Lie algebra of the $SU(1, 1)$ group.
In this work we study and exactly solve the Dirac oscillator with three different topological defects, namely the cosmic string spacetime ($Λ_\mp$), the magnetic cosmic string spacetime ($Θ_\mp$) and the cosmic dislocation spacetime ($Π_\mp$). Moreover, we show that the radial part of this problem possess an $SU(1,1)$ symmetry. From this, we obtain the wave functions and their respective energy spectrum by means of the Schrödinger factorization. Also, we compute the radial coherent states of the eigenfunctions of each problem and their time evolution.
We show that the $(2+1)$-dimensional Dirac-Moshinsky oscillator coupled to an external magnetic field can be treated algebraically with the $SU(1,1)$ group theory and its group basis. We use the $su(1,1)$ irreducible representation theory to find the energy spectrum and the eigenfunctions. Also, with the $su(1,1)$ group basis we construct the relativistic coherent states in a closed form for this problem.
A generalized definition of a deformation of the fermionic oscillator (k-fermionic oscillators) is proposed. Two prescriptions for the construction of generalized Grassmann coherent states for this kind of oscillators are derived. The two prescriptions differs in the nature of the generalized Grassmann variables used. While we use Majid's definition for such variables in the first case, Kerner's definition is used in the second one.
Two new types of coherent states associated with the C_λ-extended oscillator, where C_λ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut-Girardello and the Perelomov su(1,1) coherent states for λ=2, as well as the annihilation-operator coherent states of the C_λ-extended oscillator spectrum generating algebra for higher λvalues. The second ones, which are eigenstates of the C_λ-extended oscillator annihilation operator, extend to higher λvalues the paraboson coherent states, to which they reduce for λ=2. All these states satisfy a unity res…
We construct two commuting sets of creation and annihilation operators for the PT-symmetric oscillator. We then build coherent states of the latter as eigenstates of such annihilation operators by employing a modified version of the normalization integral that is relevant to PT-symmetric systems. We show that the coherent states are normalizable only in the range (0, 1) of the underlying coupling parameter $α$.
In this note, by using the result in one variable, we obtain asymptotic expansions of oscillatory integrals for certain multivariable phase functions with {\bf degenerate} singular points. Moreover by using this result, we have asymptotic expansions of oscillatory integrals with phase function of type $A_{k}$, $E_6$, $E_8$-function germs.
We review some basic elements on k-fermions, which are objects interpolating between bosons and fermions. In particular, we define k-fermionic coherent states and study some of their properties. The decomposition of a Q-uon into a boson and a k-fermion leads to a definition of fractional supercoherent states. Such states involve bosonic coherent states and k-fermionic coherent states. We construct an Hamiltonian which generalizes the ordinary (or Z_2-graded) supersymmetric oscillator Hamiltonian. Our Hamiltonian describes a fractional (or Z_k-graded) supersymmetric oscillator for which the fra…
Exactly-solvable model of the linear singular oscillator in the relativistic configurational space is considered. We have found wavefunctions and energy spectrum for the model under study. It is shown that they have correct non-relativistic limits.
We revisit the quantized version of the harmonic oscillator obtained through a q-dependent family of coherent states. For each q, 0< q < 1, these normalized states form an overcomplete set that resolves the unity with respect to an explicit measure. We restrict our study to the case in which 1/q is a quadratic unit Pisot number: the q-deformed integers form Fibonacci-like sequences of integers. We then examine the main characteristics of the corresponding quantum oscillator: localization in the configuration and in the phase spaces, angle operator, probability distributions and related statist…
We study the Dunkl oscillator in two dimensions by the $su(1,1)$ algebraic method. We apply the Schrödinger factorization to the radial Hamiltonian of the Dunkl oscillator to find the $su(1,1)$ Lie algebra generators. The energy spectrum is found by using the theory of unitary irreducible representations. By solving analytically the Schrödinger equation, we construct the Sturmian basis for the unitary irreducible representations of the $su(1,1)$ Lie algebra. We construct the $SU(1,1)$ Perelomov radial coherent states for this problem and compute their time evolution.
In the case of a one-dimensional nonsingular Hamiltonian $H$ and a singular supersymmetric partner $H_a$, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply this method to the singular isotonic oscillator with two singularities on the real axis. We also applied …
Two new types of coherent states associated with the $C_λ$-extended oscillator, where $C_λ$ is the cyclic group of order $λ$, are introduced. They satisfy a unity resolution relation in the $C_λ$-extended oscillator Fock space (or in some subspace thereof) and give rise to Bargmann representations of the latter, wherein the generators of the $C_λ$-extended oscillator algebra are realized as differential-operator-valued matrices.
In this paper, we first discuss the general properties of an intermediate-statistics quantum bracket, $[ u,v]_{n}=uv-e^{i2π/(n+1)}vu$, which corresponds to intermediate statistics in which the maximum occupation number of one quantum state is an arbitrary integer, $n$. A further study of the operator realization of intermediate statistics is given. We construct the intermediate-statistics coherent state. An intermediate-statistics oscillator is constructed, which returns to bosonic and fermionic oscillators respectively when $n\to \infty $ and $n=1$. The energy spectrum of such an intermediate…
In this parer, q-deformed oscillator for pseudo-Hermitian systems is investigated and pseudo-Hermitian appropriate coherent and squeezed states are studied. Also, some basic properties of these states is surveyed. The over-completeness property of the para-Grassmannian pseudo-Hermitian coherent states (PGPHCSs) examined, and also the stability of coherent and squeezed states discussed. The pseudo-Hermitian supercoherent states as the product of a pseudo-Hermitian bosonic coherent state and a para-Grassmannian pseudo-Hermitian coherent state introduced, and the method also developed to define p…
The $C_λ$-extended oscillator spectrum generating algebra is shown to be a $C_λ$-extended $(λ-1)$th-degree polynomial deformation of su(1,1). Its coherent states are constructed. Their statistical and squeezing properties are studied in detail. Such states include both some Barut-Girardello and the standard $λ$-photon coherent states as special cases.
In this work, we generate a family of quantum potentials that are non-rational extensions of the harmonic oscillator. Such a family can be obtained via two different but equivalent supersymmetric transformations. We construct ladder operators for these extensions as the product of the intertwining operators of both transformations. Then, we generate families of Barut-Girardello coherent states and analyze some of their properties as temporal stability, continuity on the label, and completeness relation. Moreover, we calculate mean-energy values, time-dependent probability densities, Wigner fun…
The solving of the Schrodinger equation for a position-dependent mass quantum system is studied in two ways. First, it is found the interaction which must be applied on a mass m(x) in order to supply it with a particular spectrum of energies. Second, given a specific potential V(x) acting on the mass m(x), the related spectrum is found. The method of solution is applied to a wide class of position-dependent mass oscillators and the corresponding coherent states are constructed. The analytical expressions of such position-dependent mass coherent states preserve the functional structure of the G…
Exactly solvable model of the quantum isotropic three-dimensional singular oscillator in the relativistic configurational $\vec r$-space is proposed. We have found the radial wavefunctions, which are expressed through the continuous dual Hahn polynomials and energy spectrum for the model under consideration. It is shown that they have the correct non-relativistic limits.
Karlin and McGregor's d-variable Hahn polynomials are shown to arise in the (d+1)-dimensional singular oscillator model as the overlap coefficients between bases associated to the separation of variables in Cartesian and hyperspherical coordinates. These polynomials in d discrete variables depend on d+1 real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution. The focus is put on the d=2 case for which the connection with the three-dimensional singular oscillator is used to derive the main properties of the polynomials: forward/backward shift operator…
In this paper we propose a symmetric q-deformed Tamm-Dancoff (S-TD) oscillator algebra and study its representation, coordinate realization, and main properties. In particular, the non-Fibonacci (more exactly, quasi-Fibonacci) nature of S-TD oscillator is established, the possibility of relating it to certain p,q-deformed oscillator family shown, the occurrence of the pairwise accidental degeneracy proven. We also find the coherent state for the S-TD oscillator and show that it satisfies completeness relation. Main advantage of the S-TD model over usual Tamm-Dancoff oscillator is that due to (…
We construct the photon added coherent states of a noncommutative harmonic oscillator associated to a $q$-deformed oscillator algebra. Various nonclassical properties of the corresponding system are explored, first, by studying two different types of higher order quadrature squeezing, namely the Hillery-type and the Hong--Mandel-type and, second, by testing the sub-Poissonian nature of photon statistics in higher order with the help of the correlation function and the Mandel parameter. By comparing our results with those of the usual harmonic oscillator, we notice that the quadratures and phot…
