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.
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Dedicated to Asim Orhan Barut (1926-1994) on the occasion of his $100^{\rm th}$ birthday. Talk presented at the GROUP36, Valladolid, Spain, July 13-17, 2026. 14 pages
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.
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
We review supersymmetry (SUSY) in nonrelativistic quantum mechanics emphasizing algebraic aspects.We discuss the Hamiltonian subgroup implementing supersymmetry as well as the corresponding algebra of the SUSY generators.In the SUSYQM framework, the two distinct partner potentials are connected by a superpotential satisfying a Riccati differential equation.A full Hamiltonian operator in an extended Hilbert space is defined in order to render the supersymmetry manifest.As a result, the eigenfunctions of the original potentials are connected by generalized ladder operators.We provide an explicit realization of the abstract supersymmetry group for SUSYQM depending on one real and two Grassmann parameters.
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.
Recently, we have observed that the bosonized supersymmetric quantum mechanics leads naturally to the R-deformed Heisenberg algebra (RDHA) related to parabosons and parafermions. Here we show that RDHA, in turn, gives rise not only to the bosonized supersymmetry associated with Witten's supersymmetric quantum mechanics but also supplies us with a supersymmetry characterized by a nonlinear superalgebra. Such nonlinear supersymmetry may be specified alternatively by a central extension of the usual N=1 superalgebra with grading operator included nontrivially as an even generator. The systems under consideration are described by the Hamiltonians of the simplest form Hn=a+a− and Ha=a−a+ and reveal supersymmetries at the special values of the deformation parameter corresponding to parabosons of odd order. The peculiar nature of parabosonic supersymmetries is encoded in essentially nonlinear structure of their supercharges realized in terms of parabosonic creation-annihilation operators $a^\\pm$. At the values of deformation parameter corresponding to the even order parabosons the spectra of the Hamiltonians Hn and Ha have no supersymmetry but reveal `holes' in comparison with bosonic spectra, whose number is correlated with the parastatistics' order. It is shown that RDHA and the associated linear and nonlinear supersymmetries are realizable in the system of two identical fermions. The nonlinear supersymmetry of pure parabosonic systems possesses the structure of a reduced parasupersymmetry and may also be realized by appropriate modification of the classical analog of Witten's supersymmetric quantum mechanics.
ly R is the Klein operator ±exp[iπ(Ĥ − E0)] while in Schrodinger coordinate representation, first investigated by Yang, R is realised by ±P where P is the parity operator: P|x >= ±|x >, P−1 = P, P = 1, PxP−1 = −x. (7) The basic (anti-)commutation relation (1) and (3) together with their derived relation (4) will be referred to here as constituting the WH algebra which is in fact a parabose algebra for one degree of freedom. We shall assume here in after, without loss of generality, that c is positive, i.e. c =| c |> 0. Thus, in coordinate representation the generalized quantization a la Wigner requires that x = x, ∞− < x < ∞, p = −i d dx + ic 2x P, R = P. (8) Indeed, following Yang representation [6] we obtain the coordinate representation for the ladder operators as given by â± −→ a±c 2 = 1 √ 2 ( ± d dx ∓ c 2x P − x ) . (9) Yang’s wave mechanical description was further investigated in [7, 8]. The Wigner-Heisenberg Algebra in Quantum Mechanics http://dx.doi.org/10.5772/55994 479 4 Advances in Quantum Mechanics The present author have applied a super-realization so that R = Σ3 to illustrate the first application of our operator method to the cases of the Hamiltonian of an isotonic oscillator (harmonic plus a centripetal barrier) system. To obtain a super-realisation of the WH algebra, we introduce, in addition to the usual bosonic coordinates (x,−i d dx ), the fermionic ones b∓(= (b±)†) that commute with the bosonic set and are represented in terms of the usual Pauli matrices Σi, (i = 1, 2, 3). Indeed, expressing a ±( c 2 ) in the following respective factorised forms: a( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( 1 2 x)( d dx )exp( −1 2 x2)x−(1/2)cΣ3 (10) a−( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( −1 2 x)( −d dx )exp( 1 2 x2)x−(1/2)cΣ3 , (11) where these lader operators satisfy the algebra of Wigner-Heisenberg. From (1), (11) and (10) the Wigner Hamiltonian becomes H( c 2 ) = 1 2 [ a( c 2 ), a−( c 2 ) ] + = ( H−( c 2 − 1) 0 0 H+( c 2 − 1) = H−( c 2 ) ) , (12) where the even and odd sector Hamiltonians are respectively given by H−( c 2 − 1) = 1 2 { − d 2 dx2 + x + 1 x2 ( c 2 )( c 2 − 1) }
We extend Witten's supersymmetry (SUSY) formulation for Hamiltonian systems to a system of annihilation operator eigenvalue equations associated with the SUSY isotonic oscillator, which, as we show, define SUSY canonical supercoherent states containing mixtures of both pure bosonic and pure fermionic counterparts. Specifically, a graded Lie algebra structure analogous to Witten's SUSY quantum mechanical algebra is realized in which only annihilation operators participate, all expressed in terms of the Wigner annihilation operator of a related super-Wigner oscillator system.