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arXiv:1201.2372 · Journal of Physics A Mathematical and Theoretical

Pseudo-Hermitian coherent states under the generalized quantum condition with position-dependent mass

Sid-Ahmed Yahiaoui, M. Bentaïba

Published Oct 23, 2012• 9 citations
arXivHTMLPDF

Abstract

In the context of the factorization method, we investigate the pseudo-Hermitian coherent states and their Hermitian counterpart coherent states under the generalized quantum condition in the framework of a position-dependent mass. By considering a specific modification in the superpotential, suitable annihilation and creation operators are constructed in order to reproduce the Hermitian counterpart Hamiltonian in the factorized form. We show that by means of these ladder operators, we can construct a wide range of exactly solvable potentials as well as their accompanying coherent states. Alternatively, we explore the relationship between the pseudo-Hermitian Hamiltonian and its Hermitian counterparts, obtained from a similarity transformation, to construct the associated pseudo-Hermitian coherent states. These latter preserve the structure of Perelomov's states and minimize the generalized position–momentum uncertainty principle. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.

DOI
10.1088/1751-8113/45/44/444034

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In the context of the factorization method, we investigate the pseudo-Hermitian coherent states and their Hermitian counterpart coherent states under the generalized quantum condition in the framework of a position-dependent mass. By considering a specific modification in the superpotential, suitable annihilation and creation operators are constructed in order to reproduce the Hermitian counterpart Hamiltonian in the factorized form. We show that by means of these ladder operators, we can construct a wide range of exactly solvable potentials as well as their accompanying coherent states. Alternatively, we explore the relationship between the pseudo-Hermitian Hamiltonian and its Hermitian counterparts, obtained from a similarity transformation, to construct the associated pseudo-Hermitian coherent states. These latter preserve the structure of Perelomov's states and minimize the generalized position–momentum uncertainty principle. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
Abstract We introduce a pseudo-squeezed bosonic ladder operator defined as a time-dependent non-Hermitian linear invariant and related to their adjoint operators via the bounded Hermitian invertible operator or metric operator. In fact, they are obtained from the squeezed transformation of the pseudo-bosons annihilation and creation operators. Thus, the pseudo-bosonic squeezed coherent states are just obtained as pseudo-displacement operator method acting on the ground pseudo-squeezed state. To our knowledge, the time-dependent pseudo-squeezed coherent states have not been constructed until now. As an illustration, we study the time-dependent non-Hermitian displaced harmonic oscillator, and the properties of these states are analysed with respect to the localization in position and to uncertainty principle.
Position-dependent e?ective mass systems are of great signi?cance due to the fact that these models have numerous applications in various areas of physics. The qualitative understanding of a complicated realistic system can be acquired by analyzing the exact solutions of a related simpli?ed model. However, the quantization of position-dependent e?ective mass systems and ?nding their solutions, involve some conceptual and mathematical di?culties of a fundamental nature. The factorization method provides us with a powerful tool for obtaining solutions and the underlying algebraic structure of the exactly solvable systems. The underlying algebra of a system has vast applications in di?erent areas of mathematics and physics, such as it plays an important role in the theory of coherent states. Coherent states are extremely useful in various areas such as quantum mechanics, quantum optics, quantum information and group theory. Inthisthesis, position-dependente?ectivemasssystemsarestudiedinthecontextoftheir quantization, ?nding the solutions, construction of the algebraic structure and associated coherent states. In the ?rst part of the thesis we mainly focus on quantizing and obtaining the exact solutions of the systems with spatially varying mass. The next part deals with the construction of the ladder operators and the inherent algebra of the pertaining systems. The associated coherent states and their properties are presented in the ?nal part of the thesis. Beside the traditional way of obtaining exact solutions by solving the Schrödinger equation there exists another elegant method to solve the systems algebraically by factorizing the corresponding Hamiltonian. This method is based on supersymmetric quantum mechanics and the integrability condition, commonly known as shape invariance. After quantizing the position-dependent e?ective mass system, this factorization technique is used to determine the energy spectrum and the corresponding wave functions. For the sake of completeness the \nmethod of solving a time-independent Schrödinger equation with spatially varying mass is also discussed. Considering a non-linear harmonic oscillator as an illustrative example, it is shown that both the above procedures produce the same results. The property of shape invariance enables us to obtain the ladder operators of the con?ning system. A general recipe for the construction of the ladder operators and inherent algebra for the position-dependent e?ective mass systems is presented. In order to exemplify the general formalism, a non-linear harmonic oscillator together with several other examples of the shape invariant systems with position-dependent e?ective mass is considered. Explicit expressions for the ladder operators and the associated algebra are presented. Usingtheladderoperatorsandtheunderlyingalgebra,thecoherentstatesforthepositiondependent e?ective mass systems are constructed and their properties are analyzed. In particular, we emphasize on various kinds of coherent states for a non-linear harmonic oscillator with spatially varying mass. By realizing SU(1,1) as the dynamic group of the system, the construction of Barut-Girardello coherent states is presented. In addition, an algebraic independent kind of coherent states, namely Gazeau-Klauder coherent states, are also constructed. The statistical properties of Barut-Girardello and Gazeau-Klauder coherent states are investigated by means of the Mandel parameter and the second order correlation function. Moreover, the temporal evolution of the Gazeau-Klauder coherent states is analysed by means of autocorrelation function. It is shown that these states mimic the phenomena of quantum revivals and fractional revivals during their time evolution
In this paper, we propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schrödinger equation. We use point canonical transformation method and obtain ground state eigenfunction of the position dependent mass Schrödinger equation. We fix the ladder operators in the deformed form and obtain explicit expression of the deformed superpotential in terms of mass distribution and its derivative. We also prove that these deformed operators lead to minimum uncertainty relations. Further, we illustrate our algorithm with two examples, in which the coherent states given for the second example are new.
By means of the unitary transformation, a new way for discussing the ordering prescription of the Schrödinger equation with a position-dependent mass (PDM) for isospectral Hamiltonian operators is presented. We show that the ambiguity parameter choices in the kinetic part of the Hamiltonian can be explained through an exact SUSY QM symmetry as well as a consequence of an accidental symmetry under the Z2 action. By making use of the unitary transformation, we construct coherent states for a family of PDM isospectral Hamiltonians from a suitable choice of ladder operators. We show that these states preserve the usual structure of Klauder-Perelomov’s states and thus saturate and minimize the position-momentum uncertainty relation (PMUR) under some special restrictions. We show that PMUR properties can be used to determine the sign of the superpotential.
In order to construct time-dependent pseudo-bosonic coherent states, first, we extend the non-Hermitian integrals of motion method to cases where the quantum systems are described by time-dependent non-Hermitian Hamiltonians; second, we introduce a pseudo-bosonic annihilation operator defined as a time-dependent non-Hermitian linear invariant. The pseudo-bosons operators are a pseudo-Hermitian extension of usual boson operators. In fact, they are obtained from the modification of usual boson commutation relations where the annihilation and creation operators are related to their adjoint operators via the bounded Hermitian invertible operator or metric operator. Thus, the pseudo-bosonic coherent states are just obtained as eigenstates of the pseudo-bosonic annihilation operator. As an illustration, we study the time-dependent non-Hermitian Swanson Hamiltonian and we compare the obtained results with those in the literature (Swanson model but time-independent).
Generalized coherent states and uncertainty relations in PT\mathcal{P}\mathcal{T}PT-symmetric position dependent mass systems
In this paper, we investigate a class of PT\mathcal{P}\mathcal{T}PT-symmetric quantum systems with position-dependent effective mass (PDEM). We factorize the PDEM Hamiltonian using a pair of first order differential operators. The resulting commutation relation introduces the notion of a position-dependent effective Planck parameter, which reduces to Planck's constant in the conventional Hermitian quantum system with constant mass. These factorization operators define a deformed phase space, within which we construct generalized coherent states to first order in the PT\mathcal{P}\mathcal{T}PT-symmetry parameter. We then revisit the Heisenberg uncertainty principle for this state and demonstrate that the deformed position and momentum operators satisfy the corresponding uncertainty relation in the PT\mathcal{P}\mathcal{T}PT-symmetric PDEM framework. Finally, we present explicit computational results for a toy-model oscillator with position-dependent effective mass in a PT\mathcal{P}\mathcal{T}PT-symmetric setting. In the conclusions section, we outline a gedankenexperiment for experimental determination of PT\mathcal{P}\mathcal{T}PT-symmetric parameter.