The technique regarding the integration within a normally ordered product of operators, which refers to the creation and annihilation operators of the harmonic oscillator coherent states, has proved to be very fruitful for different operator identities and applications in quantum optics. In this paper we propose a generalization of this technique by introducing a new operatorial approach—the diagonal ordering operation technique (DOOT)—regarding the calculations connected with the normally ordered product of generalized creation and annihilation operators that generate the generalized hypergeometric coherent states. We have pointed out a number of properties of these coherent states, including the case of mixed (thermal) states. At the same time, by particularizing the obtained results to the one-dimensional harmonic and pseudoharmonic oscillators, we get the well-known results achieved through other methods in the corresponding coherent states representation.
The technique regarding the integration within a normally ordered product of operators, which refers to the creation and annihilation operators of the harmonic oscillator coherent states, has proved to be very fruitful for different operator identities and applications in quantum optics. In this paper we propose a generalization of this technique by introducing a new operatorial approach—the diagonal ordering operation technique (DOOT)—regarding the calculations connected with the normally ordered product of generalized creation and annihilation operators that generate the generalized hypergeometric coherent states. We have pointed out a number of properties of these coherent states, including the case of mixed (thermal) states. At the same time, by particularizing the obtained results to the one-dimensional harmonic and pseudoharmonic oscillators, we get the well-known results achieved through other methods in the corresponding coherent states representation.
During the century of existence of the notion of coherent states, either linear or nonlinear, several schemes for their construction, theoretical or experimental, have been developed. Generally, the mathematical structure of coherent states depends on the choice of ladder operators, and consequently on the structure constants. In this paper, we propose a way to construct generalized coherent states for anharmonic oscillators that is based on a diagonal operator ordering technique (DOOT) applied to generalized hypergeometric functions, that is, on some of the most general special functions. These states are generated by the action of a pair of ladder operators, the creation and the annihilation, whose ordered normal product is equal to the dimensionless Hamiltonian of the quantum system. In addition, the action of these operators is easy to find if the expression for the dimensionless energy eigenvalues is known.
The generalized hypergeometric coherent states (GHCSs) have been introduced by Appl and Schiller (1). In the present paper we have extended some considerations about GHCSs to the mixed (thermal) states and applied, particularly, to the case of pseudoharmonic oscillator (PHO). The Husimi's Q distribution function and the diagonal P - distribution function, in the GHCSs representation, have been deduced for these mixed states. The obtained distribution functions were used to calculate thermal averages and to examine some nonclassical properties of the generalized hypergeometric thermal coherent states (GHTCSs), particularly for the PHO. We have also deflned and calculated the thermal analogue of the Mandel parameter and the thermal analogue of the second-order correlation function. By particularizing the parameters p and q of the hypergeometric functions, we recover the usual Barut-Girardello coherent states and their main properties for the PHO from our previous paper (9). All calculations are performed in terms of the Meijer's G-functions (2), which are related to the hypergeometric functions. This manner provides an elegance and uniformity of the obtained results and so the GHCSs become a new fleld of application for these functions. Moreover, this mathematical approach can be used also for other kind of coherent states (e.g. Klauder-Perelomov, Gazeau-Klauder or nonlinear coherent states (10), (12)).
Although the thermofield dynamics (TFD) was developed specifically for systems to which the ladder canonical operators and are associated, in the paper we showed that this approach can also be formulated for systems of deformed bosons, associated with a pair of deformed operators and . In this context we built sets of temperature dependent coherent states, both of the Barut-Girardello and of the Klauder-Perelomov type, these two types being dual to each other. In calculations involving ladder operators, both independent and temperature-dependent, we used the Diagonal Operator Ordering Technique (DOOT), which is a generalization of the Integration Within an Ordered Product technique (IWOP) applicable to canonical operators associated with the one-dimensional harmonic oscillator. We also refer to the connection between TFD and thermodynamic quantities.
In the paper we have built and examined the properties of quantum systems with nonequidistant energy levels from a point of view of a new introduced approach -the diagonal operator ordering technique (DOOT).In this frame, we examine also the properties of mixed states described by a canonical density operator.We particularize the obtained results for some particular cases (the system with Hamiltonian whose eigenfunctions are the generalized Laguerre functions, as well as the Pöschl-Teller like potentials, and the infinite quantum well).
Although the thermofield dynamics (TFD) was developed specifically for systems to which the ladder canonical operators are associated, in the paper we showed that this approach can also be formulated for systems of deformed bosons, associated with a pair of deformed operators. In this context we built sets of temperature dependent coherent states, both of the Barut-Girardello and of the Klauder-Perelomov type, these two types being dual to each other. In calculations involving ladder operators, both independent and temperature-dependent, we used the Diagonal Operator Ordering Technique (DOOT), which is a generalization of the Integration Within an Ordered Product technique (IWOP) applicable to canonical operators associated with the one-dimensional harmonic oscillator. We also refer to the connection between TFD and thermodynamic quantities.
In this paper we extend the applicability of Fox-Wright functions beyond mathematics, specifically in quantum physics. We focused our attention on a new application, on the connection between the Fox-Wright functions and the generalized coherent states formalism. We constructed the generalized coherent states in the Barut-Girardello manner, in which the Fox-Wright functions play the role of normalization functions, and we demonstrated that the Fox-Wright coherent states satisfy all general conditions imposed on the set of coherent states. In parallel, we examined the properties of both pure and mixed (thermal) Fox-Wright coherent states. All calculations were performed within the diagonal operators ordering technique (DOOT) using the Dirac's bra-ket formalism. Finally, we introduced some (specifically, integral) feedback elements that Fox-Wright coherent states induce in the theory of special functions, including a new integral representation of Fox-Wright functions.