Topic
Coherent states in mathematical physics
About: Coherent states in mathematical physics is a research topic. Over the lifetime, 732 publications have been published within this topic receiving 32024 citations.
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TL;DR: An algebraic treatment of shape-invariant potentials is discussed in this article, where an operator which reparametrizes wave functions can be related to a generalized Heisenberg-Weyl algebra.
Abstract: An algebraic treatment of shape-invariant potentials is discussed. By introducing an operator which reparametrizes wave functions, the shape-invariance condition can be related to a generalized Heisenberg-Weyl algebra. It is shown that this makes it possible to define a coherent state associated with the shape-invariant potentials.
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TL;DR: In this article, a new set of coherent states for a deformed Hamiltonian of the harmonic oscillator, called the BDS-Hamiltonian, was constructed for a new measure in the Hilbert space of the Hamiltonian eigenstates, in fact they change the inner product.
Abstract: In the paper we construct a new set of coherent states for a deformed Hamiltonian of the harmonic oscillator, previously introduced by Beckers, Debergh, and Szafraniec, which we have called the BDS-Hamiltonian. This Hamiltonian depends on the new creation operator aλ+, i.e. the usual creation operator displaced with the real quantity λ. In order to construct the coherent states, we use a new measure in the Hilbert space of the Hamiltonian eigenstates, in fact we change the inner product. This ansatz assures that the set of eigenstates be orthonormalized and complete. In the new inner product space the BDS-Hamiltonian is self-adjoint. Using these coherent states, we construct the corresponding density operator and we find the P-distribution function of the unnormalized density operator of the BDS-Hamiltonian. Also, we calculate some thermal averages related to the BDS-oscillators system which obey the quantum canonical distribution conditions.
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TL;DR: In this article, the authors generalized the nonlinear coherent states based on hypergeometric type operators associated to the Weyl-Heisenberg group to the arbitrary Lie group SU(1, 1), which admit a resolution of the identity through positive definite measures on the complex plane.
Abstract: The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.
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TL;DR: In this paper, the Laplace transform was used to derive the statistical and geometrical properties of nonlinear coherent states of the Barut-Girardello and Perelomov coherent states.
Abstract: Various aspects of coherent states of nonlinear $su(2)$ and $su(1,1)$ algebras are studied. It is shown that the nonlinear $su(1,1)$ Barut-Girardello and Perelomov coherent states are related by a Laplace transform. We then concentrate on the derivation and analysis of the statistical and geometrical properties of these states. The Berry's phase for the nonlinear coherent states is also derived.
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02 Apr 2023TL;DR: In this paper , a class of generalized coherent states for associated Jacobi polynomials and hypergeometric functions was investigated, satisfying the resolution of the identity with respect to a weight function expressed in terms of Meijer's G-function.
Abstract: In continuation of our previous works J. Phys. A: Math. Gen. 35, 9355-9365 (2002), J. Phys. A: Math. Gen. 38, 7851 (2005) and Eur. Phys. J. D 72, 172 (2018), we investigate a class of generalized coherent states for associated Jacobi polynomials and hypergeometric functions, satisfying the resolution of the identity with respect to a weight function expressed in terms of Meijer's G-function. We extend the state Hilbert space of the constructed states and discuss the property of the reproducing kernel and its analytical expansion. Further, we provide the expectation values of observables relevant to this quantum model. We also perform the quantization of the complex plane, compute and analyze the probability density and the temporal stability in these states. Using the completeness relation provided by the coherent states, we achieve the thermodynamic analysis in the diagonal $P$-representation of the density operator.