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

Symmetry

Supersymmetric Quesne-Dunkl Quantum Mechanics on Radial Lines

Fethi Bouzeffour

Published Nov 10, 2024• 4 citations
Publisher

Abstract

Quantum deformations offer valuable perspectives into quantum mechanics, particularly by advancing our understanding of symmetry and algebraic structures.The Dunkl oscillator, which integrates Dunkl operators into the harmonic oscillator framework, advances the system’s algebraic properties and opens new approaches for exploring quantum phenomena. Supersymmetric quantum mechanics (SSQM) unifies bosonic and fermionic aspects and facilitates the construction of solvable models using generalized Dunkl operators. This paper introduces a new approach to the Dunkl oscillator, employing a complex reflection operator to deepen the understanding of its connection to Hermite polynomials on radial lines. The results offer new perspectives on the Dunkl oscillator’s algebraic structure and its relevance to SSQM and quantum deformation theory, expanding the potential for discovering solvable quantum models.

DOI
10.3390/sym16111508

Related papers

Unified supersymmetric transformations for the harmonic oscillator and its rational extension

GÉZA LÉVAI, Tibor Soltész

Abstract We discussed transformations of supersymmetric quantum mechanics by considering a general six-parameter function, from which the superpotential and the supersymmetric partner potentials V − ( r ) and V + ( r ) were constructed. By specific choice of the parameters, V − ( r ) reproduced the radial harmonic oscillator, or its rationally extended version, with solutions containing X 1 -type Laguerre polynomials. The choice of parameters also determined which of the four possible SUSY transformations T i , i = 1, ...4 were employed. Within this formalism, the relation between the radial harmonic oscillator and its rational extension can be discussed in a straightforward way, furthermore, several aspects of SUSYQM and exactly solvable potentials can be demonstrated in a unified framework on a level suitable for basic quantum mechanics courses.

European Journal of Physics•Dec 6, 2019•5 citations

Hidden Supersymmetry in Wigner-Yang Quantum Mechanics

Georg Junker

We consider a quantum system on the real line obeying a deformed Heisenberg algebra originally proposed by Wigner in 1950. Its explicit coordinate representation was provided by Yang in 1951 and in essence is identical in form with Dunkl's difference-differential operator introduced in 1989 in connection with roots systems of refection groups. Under certain conditions such quantum systems exhibit a supersymmetric (SUSY) structure where the reflection operator acts as the grading operator. We present a generalisation of Yang's representation by first considering only of one the two equations of motion in phase space. The corresponding non-interacting system is found to represent Witten's model of SUSY quantum mechanics. Imposing also the second equation of motion the original result of Wigner and Yang is reconsidered by extending their discussion to general symmetric potentials on the real line. As explicit example we discuss the harmonic oscillator and an attractive Coulomb-like potential $V(x)=-γ/|x|$. We also establish a Hooke-Newton duality between this Coulomb-like system and the original Wigner-Yang harmonic oscillator system.

math-phCross-listedarXiv:2608.04503•Aug 6, 2026

Testing $(q)$-Deformed Dunkl-Fokker-Planck Equation Algebra with Supersymmetry (SUSY) and Foldy-Wouthuysen (FW) Measurement

Abdelmalek Bouzenada

In this study, a relativistic formulation of the $(q)$-deformed Dunkl-Fokker-Planck equation in $(1+1)$-dimensions is constructed within the reflection-deformed quantum framework. In this case, the formalism includes $(q)$-deformed Dunkl operators and reflection symmetry to build a generalized dynamical structure for a relativistic quantum systems framework. Moreover, the corresponding $(q)$-Wigner-Dunkl supersymmetric configuration is established via the construction of deformed ladder operators and supersymmetric algebraic relations, yielding a consistent spectral representation of the model within the algebraic framework. The analysis extends to the harmonic oscillator with centrifugal interaction, where exact algebraic solutions, similarity reduction techniques, and closed energy spectra are obtained analytically in detail. The role of the deformation parameter and reflection operator on spectral properties and wavefunction structure is examined systematically in detail. A generalized Foldy-Wouthuysen (FW) transformation is introduced within the deformed Dunkl framework to achieve relativistic decoupling of positive- and negative-energy sectors within the present theoretical formulation. In this case, this approach yields an effective reduced Hamiltonian, including higher-order relativistic and deformation-induced terms. Also, the associated Dunkl-Fokker-Planck dynamics generated through high-order FW reduction are examined in detail for reflection-deformed relativistic quantum systems. In this context, results obtained here yield a unified algebraic and relativistic description of $(q)$-deformed Dunkl structures and construct a consistent framework for investigating supersymmetric and relativistic properties in reflection-symmetric quantum models in general.

arXiv:2605.13938•May 13, 2026•0 citations

Supersymmetric Wigner–Dunkl quantum mechanics

Shi‐Hai Dong, Won Sang Chung, Georg Junker +1 more

Wigner–Dunkl quantum mechanics can be viewed as deformed quantum mechanics, where the commutator between canonical momentum and position operators contains in addition a reflection operator. In the present work we carry over the general concepts of supersymmetric (SUSY) quantum mechanics to the Wigner–Dunkl quantum formulation. We investigate the SUSY ground states, the SUSY transformations and introduce a generalized shape invariance allowing for explicit solutions. In this way, exact solutions for a certain class of SUSY potentials can be provided. As example the harmonic oscillator is considered. We also take a look into a singular oscillator model, which does not belong to the shape-invariance class. Nevertheless, exact solutions can be provided, showing the peculiarities of that model in the SUSY Wigner–Dunkl quantum mechanics. A modification of that singular model is presented which restores shape invariance

Results in Physics•Jun 3, 2022•25 citations

q-deformation of radial problems: the simple harmonic oscillator in two dimensions

Peter Jarvis, Timothy H. Baker

The possibility of q-deformation of known exactly solvable quantum mechanical models is considered via the dynamical symmetry of the radial problem. The algebraic structure of the D-dimensional simple harmonic oscillator is analysed from this point of view. In the two-dimensional case, q-deformed bosons are introduced for the radial degrees of freedom, which together with the standard angular operators yield the so(2)(+)so q (2,1) dynamical algebra. The corresponding Hilbert space is identified and possible q-deformed Hamiltonians described.

Journal of Physics A Mathematical and General•Feb 21, 1993•4 citations

su(1,1) Symmetry and Exact Solutions of the Dunkl-Klein-Gordon Equation in Higher Dimensions

M. Salazar-Ramírez, B. C. Lütfüoğlu, Jesús Alfredo Martínez-Nuño

We investigate the $d$-dimensional Dunkl--Klein--Gordon equation for a scalar particle within an algebraic framework. By employing Schrödinger factorization, we construct the generators of the $\mathfrak{su}(1,1)$ algebra and establish the associated symmetry of the radial sector. The corresponding energy spectra are derived from unitary irreducible representations, and the Sturmian radial bases are obtained analytically. We analyze the $d$-dimensional Dunkl--Klein--Gordon oscillator and the bound-state sector of the $d$-dimensional Dunkl--Klein--Gordon equation with a Dunkl--Coulomb-like potential. Furthermore, $\mathrm{SU}(1,1)$ coherent states are constructed and their time evolution is analyzed, revealing a characteristic radial oscillation behavior. The results show that the Dunkl deformation modifies the spatial and spectral structure of the system and, in the oscillator sector, introduces explicit parity-dependent energy shifts while preserving the underlying radial algebraic dynamics.

quant-pharXiv:2606.27589•Physica Scripta•Jun 29, 2026•0 citations
Quantum deformations offer valuable perspectives into quantum mechanics, particularly by advancing our understanding of symmetry and algebraic structures.The Dunkl oscillator, which integrates Dunkl operators into the harmonic oscillator framework, advances the system’s algebraic properties and opens new approaches for exploring quantum phenomena. Supersymmetric quantum mechanics (SSQM) unifies bosonic and fermionic aspects and facilitates the construction of solvable models using generalized Dunkl operators. This paper introduces a new approach to the Dunkl oscillator, employing a complex reflection operator to deepen the understanding of its connection to Hermite polynomials on radial lines. The results offer new perspectives on the Dunkl oscillator’s algebraic structure and its relevance to SSQM and quantum deformation theory, expanding the potential for discovering solvable quantum models.
Abstract We discussed transformations of supersymmetric quantum mechanics by considering a general six-parameter function, from which the superpotential and the supersymmetric partner potentials V − ( r ) and V + ( r ) were constructed. By specific choice of the parameters, V − ( r ) reproduced the radial harmonic oscillator, or its rationally extended version, with solutions containing X 1 -type Laguerre polynomials. The choice of parameters also determined which of the four possible SUSY transformations T i , i = 1, ...4 were employed. Within this formalism, the relation between the radial harmonic oscillator and its rational extension can be discussed in a straightforward way, furthermore, several aspects of SUSYQM and exactly solvable potentials can be demonstrated in a unified framework on a level suitable for basic quantum mechanics courses.
We consider a quantum system on the real line obeying a deformed Heisenberg algebra originally proposed by Wigner in 1950. Its explicit coordinate representation was provided by Yang in 1951 and in essence is identical in form with Dunkl's difference-differential operator introduced in 1989 in connection with roots systems of refection groups. Under certain conditions such quantum systems exhibit a supersymmetric (SUSY) structure where the reflection operator acts as the grading operator. We present a generalisation of Yang's representation by first considering only of one the two equations of motion in phase space. The corresponding non-interacting system is found to represent Witten's model of SUSY quantum mechanics. Imposing also the second equation of motion the original result of Wigner and Yang is reconsidered by extending their discussion to general symmetric potentials on the real line. As explicit example we discuss the harmonic oscillator and an attractive Coulomb-like potential V(x)=−γ/∣x∣V(x)=-γ/|x|V(x)=−γ/∣x∣. We also establish a Hooke-Newton duality between this Coulomb-like system and the original Wigner-Yang harmonic oscillator system.
Testing (q)(q)(q)-Deformed Dunkl-Fokker-Planck Equation Algebra with Supersymmetry (SUSY) and Foldy-Wouthuysen (FW) Measurement
In this study, a relativistic formulation of the (q)(q)(q)-deformed Dunkl-Fokker-Planck equation in (1+1)(1+1)(1+1)-dimensions is constructed within the reflection-deformed quantum framework. In this case, the formalism includes (q)(q)(q)-deformed Dunkl operators and reflection symmetry to build a generalized dynamical structure for a relativistic quantum systems framework. Moreover, the corresponding (q)(q)(q)-Wigner-Dunkl supersymmetric configuration is established via the construction of deformed ladder operators and supersymmetric algebraic relations, yielding a consistent spectral representation of the model within the algebraic framework. The analysis extends to the harmonic oscillator with centrifugal interaction, where exact algebraic solutions, similarity reduction techniques, and closed energy spectra are obtained analytically in detail. The role of the deformation parameter and reflection operator on spectral properties and wavefunction structure is examined systematically in detail. A generalized Foldy-Wouthuysen (FW) transformation is introduced within the deformed Dunkl framework to achieve relativistic decoupling of positive- and negative-energy sectors within the present theoretical formulation. In this case, this approach yields an effective reduced Hamiltonian, including higher-order relativistic and deformation-induced terms. Also, the associated Dunkl-Fokker-Planck dynamics generated through high-order FW reduction are examined in detail for reflection-deformed relativistic quantum systems. In this context, results obtained here yield a unified algebraic and relativistic description of (q)(q)(q)-deformed Dunkl structures and construct a consistent framework for investigating supersymmetric and relativistic properties in reflection-symmetric quantum models in general.
Wigner–Dunkl quantum mechanics can be viewed as deformed quantum mechanics, where the commutator between canonical momentum and position operators contains in addition a reflection operator. In the present work we carry over the general concepts of supersymmetric (SUSY) quantum mechanics to the Wigner–Dunkl quantum formulation. We investigate the SUSY ground states, the SUSY transformations and introduce a generalized shape invariance allowing for explicit solutions. In this way, exact solutions for a certain class of SUSY potentials can be provided. As example the harmonic oscillator is considered. We also take a look into a singular oscillator model, which does not belong to the shape-invariance class. Nevertheless, exact solutions can be provided, showing the peculiarities of that model in the SUSY Wigner–Dunkl quantum mechanics. A modification of that singular model is presented which restores shape invariance
The possibility of q-deformation of known exactly solvable quantum mechanical models is considered via the dynamical symmetry of the radial problem. The algebraic structure of the D-dimensional simple harmonic oscillator is analysed from this point of view. In the two-dimensional case, q-deformed bosons are introduced for the radial degrees of freedom, which together with the standard angular operators yield the so(2)(+)so q (2,1) dynamical algebra. The corresponding Hilbert space is identified and possible q-deformed Hamiltonians described.
We investigate the ddd-dimensional Dunkl--Klein--Gordon equation for a scalar particle within an algebraic framework. By employing Schrödinger factorization, we construct the generators of the su(1,1)\mathfrak{su}(1,1)su(1,1) algebra and establish the associated symmetry of the radial sector. The corresponding energy spectra are derived from unitary irreducible representations, and the Sturmian radial bases are obtained analytically. We analyze the ddd-dimensional Dunkl--Klein--Gordon oscillator and the bound-state sector of the ddd-dimensional Dunkl--Klein--Gordon equation with a Dunkl--Coulomb-like potential. Furthermore, SU(1,1)\mathrm{SU}(1,1)SU(1,1) coherent states are constructed and their time evolution is analyzed, revealing a characteristic radial oscillation behavior. The results show that the Dunkl deformation modifies the spatial and spectral structure of the system and, in the oscillator sector, introduces explicit parity-dependent energy shifts while preserving the underlying radial algebraic dynamics.