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:hep-th/9309160 · International Journal of Modern Physics A

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

Ursula Carow-Watamura, Satoshi Watamura

Published Sep 10, 1994• 17 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0217751x94001618

Related papers

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 Quantum Group as a Symmetry

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.

Progress of Theoretical Physics Supplement•Jan 1, 1995•2 citations

Harmonic oscillators in q implicate-order theory and order structures

Zai-Zhe Zhong

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.

Physical Review A•May 1, 1997•0 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 mechanical Schrodinger picture of a q-oscillator

Trong Tuong Truong

A q-deformed version of standard quantum mechanics in the coordinate Schrodinger picture is obtained by replacing the ordinary coordinate derivative by the so-called q-discrete derivative as the representative of the momentum operator. The chosen q-discrete derivative is symmetric with respect to the exchange of q and q -1 . Under the usually adopted assumptions a q-deformed Schrodinger equation is derived for a harmonic oscillator. The complete set of eigenfunctions can be explicitly constructed as special q-functions and the corresponding energy eigenvalues are identical to those obtained by Biedenharn in his pioneering work (1989). This q-deformed oscillator exhibits a rich novel structure including dynamical symmetry and in the limit q to 1 it reveals some hitherto unknown features of the harmonic oscillator eigenfunctions.

Journal of Physics A Mathematical and General•Jun 7, 1994•10 citations

THE q-SCHRÖDINGER EQUATION

Joseph A. Minahan

We consider the Schrödinger equation for the one-dimensional harmonic oscillator, but with the normal derivative replaced by a q-derivative. The normalizable solutions are found and the q-generalization of the Hermite polynomials is given. The free equation is also considered, but no normalizable eigenstates exist even if the system is in a box.

Modern Physics Letters A•Dec 20, 1990•148 citations
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.
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.
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.
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.
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.
A q-deformed version of standard quantum mechanics in the coordinate Schrodinger picture is obtained by replacing the ordinary coordinate derivative by the so-called q-discrete derivative as the representative of the momentum operator. The chosen q-discrete derivative is symmetric with respect to the exchange of q and q -1 . Under the usually adopted assumptions a q-deformed Schrodinger equation is derived for a harmonic oscillator. The complete set of eigenfunctions can be explicitly constructed as special q-functions and the corresponding energy eigenvalues are identical to those obtained by Biedenharn in his pioneering work (1989). This q-deformed oscillator exhibits a rich novel structure including dynamical symmetry and in the limit q to 1 it reveals some hitherto unknown features of the harmonic oscillator eigenfunctions.