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

Physical Review A

Harmonic oscillators in q implicate-order theory and order structures

Zai-Zhe Zhong

Published May 1, 1997• 0 citations
Publisher

Abstract

In this paper, the q Euclidean space and the q deformed harmonic oscillators in this space are described by an algebraic system. In this system we can define the raising (creation) and the lowering (annihilation) operators. These operators form a quantum hyperplane and the corresponding covariant q partial derivatives. From the viewpoint of this quantum hyperplane, the operator q Schr\"odinger equation and the explanation of its solutions are discussed. In addition, some related q implicate-order structures are presented.

DOI
10.1103/physreva.55.3341

Related papers

THE q-DEFORMED SCHRÖDINGER EQUATION OF THE HARMONIC OSCILLATOR ON THE QUANTUM EUCLIDEAN SPACE

Ursula Carow-Watamura, Satoshi Watamura

We consider the q-deformed Schrödinger equation of the harmonic oscillator on the N-dimensional quantum Euclidean space. The creation and annihilation operators are found, which systematically produce all energy levels and eigenfunctions of the Schrödinger equation. In order to get the q series representation of the eigenfunction, we also give an alternative way to solve the Schrödinger equation which is based on the q analysis. We represent the Schrödinger equation by the q difference equation and solve it by using q polynomials and q exponential functions.

arXiv:hep-th/9309160•International Journal of Modern Physics A•Sep 10, 1994•17 citations

New Deformation of quantum oscillator algebra: Representation and some application

Sama Arjika, Dine Ousmane Samary, Ezinvi Baloïtcha +1 more

This work addresses the study of the oscillator algebra, defined by four parameters $p$, $q$, $α$, and $ν$. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.

arXiv:1210.7259•Oct 26, 2012•0 citations

The Quantum Group as a Symmetry - The Schrödinger equation of the $N$-dimensional $q$-deformed Harmonic Oscillator -

Ursula Carow-Watamura, Satoshi Watamura

With the aim to construct a dynamical model with quantum group symmetry, the $q$-deformed Schrödinger equation of the harmonic oscillator on the $N$-dimensional quantum Euclidian space is investigated. After reviewing the differential calculus on the $q$-Euclidian space, the $q$-analog of the creation-annihilation operator is constructed. It is shown that it produces systematically all eigenfunctions of the Schrödinger equation and eigenvalues. We also present an alternative way to solve the Schrödinger equation which is based on the $q$-analysis. We represent the Schrödinger equation by the $q$-difference equation and solve it by using $q$-polynomials and $q$-exponential functions. The problem of the involution corresponding to the reality condition is discussed.

arXiv:hep-th/9409040•Sep 8, 1994•1 citation

The q -Deformation of the Superoscillators and Their q -Supercoherent States

Hao San-Ru, Wang LuYa

We discuss the superoscillators in thc supersymmetric quantum systems, and give the solution to the system. By making use of the annihilation and creation operators and their q -deformed operators, we also give the q -deformed quantum harmonic oscillator model in the supersymmetric quantum systems. The q -supercoherent states associated with the q -deformed supersymmetric quantum harmonic oscillators are constructed explicitly, and their properties are investigated. The uncertainty relations for q -supercoherent states are discussed as well.

Communications in Theoretical Physics•Oct 30, 2000•0 citations

Partial algebraization and a q-deformed harmonic oscillator

Abilio De Freitas, S. Salamó

From the algebraic treatment of the quasi-solvable systems, and a q-deformation of the associated $su(2)$ algebra, we obtain exact solutions for the q-deformed Schrodinger equation with a 3-dimensional q-deformed harmonic oscillator potential.

arXiv:hep-th/9908098•Aug 13, 1999•0 citations

Factorization and q-Deformed Algebra of Quantum Anharmonic Oscillator

Dongsu Bak A, S P Kim, S K Kim +2 more

We have studied the underlying algebraic structure of the anharmonic oscillator by using the variational perturbation theory. To the first order of the variational perturbation, the Hamiltonian is found to be factorized into a supersymmetric form in terms of the annihilation and creation operators, which satisfy a q-deformed algebra. This algebraic structure is used to construct all the eigenstates of the Hamiltonian.

arXiv:hep-th/9808144•Journal of the Korean Physical Society•Aug 24, 1998•2 citations
In this paper, the q Euclidean space and the q deformed harmonic oscillators in this space are described by an algebraic system. In this system we can define the raising (creation) and the lowering (annihilation) operators. These operators form a quantum hyperplane and the corresponding covariant q partial derivatives. From the viewpoint of this quantum hyperplane, the operator q Schr"odinger equation and the explanation of its solutions are discussed. In addition, some related q implicate-order structures are presented.
We consider the q-deformed Schrödinger equation of the harmonic oscillator on the N-dimensional quantum Euclidean space. The creation and annihilation operators are found, which systematically produce all energy levels and eigenfunctions of the Schrödinger equation. In order to get the q series representation of the eigenfunction, we also give an alternative way to solve the Schrödinger equation which is based on the q analysis. We represent the Schrödinger equation by the q difference equation and solve it by using q polynomials and q exponential functions.
This work addresses the study of the oscillator algebra, defined by four parameters ppp, qqq, ααα, and ννν. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.
The Quantum Group as a Symmetry - The Schrödinger equation of the NNN-dimensional qqq-deformed Harmonic Oscillator -
With the aim to construct a dynamical model with quantum group symmetry, the qqq-deformed Schrödinger equation of the harmonic oscillator on the NNN-dimensional quantum Euclidian space is investigated. After reviewing the differential calculus on the qqq-Euclidian space, the qqq-analog of the creation-annihilation operator is constructed. It is shown that it produces systematically all eigenfunctions of the Schrödinger equation and eigenvalues. We also present an alternative way to solve the Schrödinger equation which is based on the qqq-analysis. We represent the Schrödinger equation by the qqq-difference equation and solve it by using qqq-polynomials and qqq-exponential functions. The problem of the involution corresponding to the reality condition is discussed.
We discuss the superoscillators in thc supersymmetric quantum systems, and give the solution to the system. By making use of the annihilation and creation operators and their q -deformed operators, we also give the q -deformed quantum harmonic oscillator model in the supersymmetric quantum systems. The q -supercoherent states associated with the q -deformed supersymmetric quantum harmonic oscillators are constructed explicitly, and their properties are investigated. The uncertainty relations for q -supercoherent states are discussed as well.
From the algebraic treatment of the quasi-solvable systems, and a q-deformation of the associated su(2)su(2)su(2) algebra, we obtain exact solutions for the q-deformed Schrodinger equation with a 3-dimensional q-deformed harmonic oscillator potential.