# On quantized Liénard oscillator and momentum dependent mass

20 Jan 2015-Journal of Mathematical Physics (American Institute of Physics Inc.)-Vol. 56, Iss: 1, pp 012105

TL;DR: In this paper, the analytical structure of the nonlinear Lienard oscillator was examined and it was shown that it is a bi-Hamiltonian system depending upon the choice of the coupling parameters.

Abstract: We examine the analytical structure of the nonlinear Lienard oscillator and show that it is a bi-Hamiltonian system depending upon the choice of the coupling parameters. While one has been recently studied in the context of a quantized momentum-dependent mass system, the other Hamiltonian also reflects a similar feature in the mass function and also depicts an isotonic character. We solve for such a Hamiltonian and give the complete solution in terms of a confluent hypergeometric function.

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TL;DR: In this paper, the construction of nonstandard Lagrangians and Hamiltonian structures for Lienard equations satisfying Chiellini condition is presented and their connection to time-dependent Hamiltonian formalism is discussed.

Abstract: The construction of nonstandard Lagrangians and Hamiltonian structures for Lienard equations satisfying Chiellini condition is presented and their connection to time-dependent Hamiltonian formalism...

37 citations

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CINVESTAV

^{1}TL;DR: The present work is intended to be complementary to other papers of the same nature and subject in current circulation to clarify concepts and notions, including some passages of the history of science, with the aim of facilitating the subject for nonspecialists.

Abstract: A short review of the main properties of coherent and squeezed states is given in the introductory form. The efforts are addressed to clarify concepts and notions, including some passages of the history of science, with the aim of facilitating the subject for nonspecialists. In this sense, the present work is intended to be complementary to other papers of the same nature and subject in current circulation.

26 citations

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TL;DR: In this paper, the authors present the quantum and classical mechanics formalisms for a particle with position-dependent mass in the context of a deformed algebraic structure, motivated by the Kappa-statistics.

Abstract: We present the quantum and classical mechanics formalisms for a particle with position-dependent mass in the context of a deformed algebraic structure (named $\kappa$-algebra), motivated by the Kappa-statistics. From this structure we obtain deformed versions of the position and momentum operators, which allow to define a point canonical transformation that maps a particle with constant mass in a deformed space into a particle with position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews-Lakshmanan oscillator, exhibiting uncertainty relations depending on the deformation.

13 citations

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TL;DR: In this paper, a particle with a position-dependent mass in the context of a deformed algebraic structure (called κ-algebra), motivated by the Kappa-statistics, is presented.

Abstract: We present the quantum and classical mechanics formalisms for a particle with a position-dependent mass in the context of a deformed algebraic structure (named κ-algebra), motivated by the Kappa-statistics. From this structure, we obtain deformed versions of the position and momentum operators, which allow us to define a point canonical transformation that maps a particle with a constant mass in a deformed space into a particle with a position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews–Lakshmanan oscillator, exhibiting uncertainty relations depending on the deformation.

12 citations

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TL;DR: In this paper, the authors explore certain classes of branched Hamiltonians in the context of nonlinear autonomous differential equation of Lienard type and two eligible elementary nonlinear models that emerge are shown to admit a feasible quantization along these lines.

Abstract: One of the less well-understood ambiguities of quantization is emphasized to result from the presence of higher-order time derivatives in the Lagrangians resulting in multiple-valued Hamiltonians. We explore certain classes of branched Hamiltonians in the context of nonlinear autonomous differential equation of Lienard type. Two eligible elementary nonlinear models that emerge are shown to admit a feasible quantization along these lines.

8 citations

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497 citations

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TL;DR: In this paper, the quantum-mechanical system which consists of a particle in one dimension subjected to a Coulomb attraction (the one-dimensional hydrogen atom) is shown to have a ground state of infinite binding energy, all the excited bound states of the system having a twofold degeneracy.

Abstract: The quantum-mechanical system which consists of a particle in one dimension subjected to a Coulomb attraction (the one-dimensional hydrogen atom) is shown to have a ground state of infinite binding energy, all the excited bound states of the system having a twofold degeneracy. The breakdown of the theorem that a one-dimensional system cannot have degeneracy is examined. The treatment illustrates a number of properties common to the quantum mechanics of one-dimensional systems.

455 citations

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TL;DR: In this paper, the spin response function for electrons confined in a quantum dot is studied within the time-dependent local spin density approximation (TDLSDA) of density functional theory.

Abstract: The spin response function for electrons confined in a quantum dot is studied within the time-dependent local spin density approximation (TDLSDA) of density functional theory. In the long-wavelength regime we predict the existence of a low-energy collective dipole (l = 1) spin mode. The evolution with electron number of the spin response is studied and compared with that of the density response. Predictions for the static dipole polarizability are given.

332 citations

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TL;DR: In this article, a finite-range density functional was used to investigate the energy and structural properties of mixed helium clusters and the possibility of doping the cluster with a molecule of sulfur hexafluoride is also considered.

Abstract: Using a finite-range density functional, we have investigated the energetics and structural features of mixed helium clusters. The possibility of doping the cluster with a molecule of sulfur hexafluoride is also considered. It is seen that the repulsion introduced by the impurity strongly modifies the properties of the smallest drops. Although only a qualitative comparison is possible, the gross features displayed by our calculations are in agreement with recent experimental findings. {copyright} {ital 1997} {ital The American Physical Society}

293 citations

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TL;DR: In this paper, a simple method of constructing potentials which is related to the work of Bhattacharjie and Sudarshan (1962) and for which the Schrodinger equation can be solved in terms of known special functions was investigated.

Abstract: The author investigates a simple method of constructing potentials which is related to the work of Bhattacharjie and Sudarshan (1962) and for which the Schrodinger equation can be solved in terms of known special functions. It turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions, satisfying linear homogeneous second-order differential equations, can be solutions of the Schrodinger equation with potentials of the form V(x)=W2(x)-W'(x). The author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape-invariant potentials.

258 citations