Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2208.10336

Generalized coherent states and uncertainty relations in $\mathcal{P}\mathcal{T}$-symmetric position dependent mass systems

Pinaki Patra

Published Aug 19, 2022• 0 citations
arXivHTMLPDF

Abstract

In this paper, we investigate a class of $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}\mathcal{T}$-symmetric PDEM framework. Finally, we present explicit computational results for a toy-model oscillator with position-dependent effective mass in a $\mathcal{P}\mathcal{T}$-symmetric setting. In the conclusions section, we outline a gedankenexperiment for experimental determination of $\mathcal{P}\mathcal{T}$-symmetric parameter.

DOI
10.48550/arxiv.2208.10336

Related papers

New $\mathcal {SU}(1,1)$ position-dependent effective mass coherent states for a generalized shifted harmonic oscillator

Sid-Ahmed Yahiaoui, M. Bentaïba

A new position-dependent effective mass coherent states (PDEM CS) related to the shifted harmonic oscillator (SHO) are deduced. This is accomplished by applying a similarity transformation to the generally deformed oscillator algebra (GDOA) generators for PDEM systems and a new set of operators that close the Lie algebra are constructed, being the PDEM CS of the basis for its unitary irreducible representation. From the Lie algebra generators, we evaluate the uncertainty relationship for a position and momentum-like operators in the PDEM CS and show that it is minimized in the sense of Barut–Girardello CS. We prove that the deduced PDEM CS preserve the same analytical form than those of Glauber states. As an illustration of our procedure, we depicted the 2D-probability density in the PDEM CS for SHO with the explicit form of the mass distribution with no singularities.

arXiv:1306.0197•Journal of Physics A Mathematical and Theoretical•Dec 17, 2013•11 citations

Supersymmetric quantum mechanics and coherent states for a deformed oscillator with position-dependent effective mass

Bruno G. da Costa, Genilson A. C. da Silva, Ignacio S. Gomez

We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator associated with a position-dependent effective mass by means of the supersymmetric formalism. From the supersymmetric partner Hamiltonians and the shape invariance technique, we obtain the eigenstates and the eigenvalues along with the ladders operators, thus showing a preservation of the supersymmetric structure in terms of the deformed counterpartners. The deformed space in supersymmetry allows to characterize position-dependent effective mass and uniform field interactions and to obtain a generalized uncertainty relation (GUP) that behaves as a distinguishability measure for the coherent states, these latter satisfying a periodic evolution for the corrections of the GUP.

arXiv:2106.08467•Journal of Mathematical Physics•Sep 1, 2021•16 citations

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

Sid-Ahmed Yahiaoui, M. Bentaïba

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'.

arXiv:1201.2372•Journal of Physics A Mathematical and Theoretical•Oct 23, 2012•9 citations

Exact solution and coherent states of an asymmetric oscillator with position-dependent mass

Bruno G. da Costa, Ignacio S. Gomez, Biswanath Rath

We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 62, 092101 (2021)] in the classical and quantum formalisms by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space (x, p) to a deformed one (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.

arXiv:2302.02172•Journal of Mathematical Physics•Jan 1, 2023•14 citations

Time-dependent q-deformed bi-coherent states for generalized uncertainty relations

Laure Gouba

We consider the time-dependent bi-coherent states that are essentially the Gazeau-Klauder coherent states for the two dimensional noncommutative harmonic oscillator. Starting from some q-deformations of the oscillator algebra for which the entire deformed Fock space can be constructed explicitly, we define the q-deformed bi-coherent states. We verify the generalized Heisenberg’s uncertainty relations projected onto these states. For the initial value in time, the states are shown to satisfy a generalized version of Heisenberg’s uncertainty relations. For the initial value in time and for the parameter of noncommutativity θ = 0, the inequalities are saturated for the simultaneous measurement of the position-momentum observables. When the time evolves, the uncertainty products are different from their values at the initial time and do not always respect the generalized uncertainty relations.

arXiv:1504.00583•Journal of Mathematical Physics•Jul 1, 2015•6 citations

Isospectral Hamiltonian for position-dependent mass for an arbitrary quantum system and coherent states

Sid-Ahmed Yahiaoui, M. Bentaïba

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.

arXiv:1611.05394•Journal of Mathematical Physics•Jun 1, 2017•8 citations
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.
New SU(1,1)\mathcal {SU}(1,1)SU(1,1) position-dependent effective mass coherent states for a generalized shifted harmonic oscillator
A new position-dependent effective mass coherent states (PDEM CS) related to the shifted harmonic oscillator (SHO) are deduced. This is accomplished by applying a similarity transformation to the generally deformed oscillator algebra (GDOA) generators for PDEM systems and a new set of operators that close the Lie algebra are constructed, being the PDEM CS of the basis for its unitary irreducible representation. From the Lie algebra generators, we evaluate the uncertainty relationship for a position and momentum-like operators in the PDEM CS and show that it is minimized in the sense of Barut–Girardello CS. We prove that the deduced PDEM CS preserve the same analytical form than those of Glauber states. As an illustration of our procedure, we depicted the 2D-probability density in the PDEM CS for SHO with the explicit form of the mass distribution with no singularities.
We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator associated with a position-dependent effective mass by means of the supersymmetric formalism. From the supersymmetric partner Hamiltonians and the shape invariance technique, we obtain the eigenstates and the eigenvalues along with the ladders operators, thus showing a preservation of the supersymmetric structure in terms of the deformed counterpartners. The deformed space in supersymmetry allows to characterize position-dependent effective mass and uniform field interactions and to obtain a generalized uncertainty relation (GUP) that behaves as a distinguishability measure for the coherent states, these latter satisfying a periodic evolution for the corrections of the GUP.
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'.
We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 62, 092101 (2021)] in the classical and quantum formalisms by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space (x, p) to a deformed one (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.
We consider the time-dependent bi-coherent states that are essentially the Gazeau-Klauder coherent states for the two dimensional noncommutative harmonic oscillator. Starting from some q-deformations of the oscillator algebra for which the entire deformed Fock space can be constructed explicitly, we define the q-deformed bi-coherent states. We verify the generalized Heisenberg’s uncertainty relations projected onto these states. For the initial value in time, the states are shown to satisfy a generalized version of Heisenberg’s uncertainty relations. For the initial value in time and for the parameter of noncommutativity θ = 0, the inequalities are saturated for the simultaneous measurement of the position-momentum observables. When the time evolves, the uncertainty products are different from their values at the initial time and do not always respect the generalized uncertainty relations.
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.