Peter Henselder
Deforming the algebraic structure of geometric algebra on the phase space with a Moyal product leads naturally to supersymmetric quantum mechanics in the star product formalism.
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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, uniform field interactions and to ob…
A simple version of the q-deformed calculus is used to generate a pair of q-nonlocal, second-order difference operators by means of deformed counterparts of Darboux intertwining operators for zero factorization energy. These deformed non-local operators may be considered as supersymmetric partners and their structure contains contributions originating in both the Hermite operator and the quantum harmonic oscillator operator. There are also extra $\pm x$ contributions. The undeformed limit, in which all q-nonlocalities wash out, corresponds to the usual supersymmetric pair of quantum mechanical…
In this study, we focus on investigating the exact relativistic bound state spectra for supersymmetric, PT-supersymmetric and non-Hermitian versions of q-deformed parameter Hulthen potential. The Hamiltonian hierarchy mechanism, namely the factorization method, is adopted within the framework of SUSYQM. This algebraic approach is used in solving of the Klein-Gordon equation with the potential cases. The results obtained analytically by executing the straightforward calculations are in consistent forms for certain values of q. Achieving the results may have a particular interest for such applic…
Uncertainty Relations and Entanglement for PQ-deformed Supersymmetric Coherent States
本文提出了具有双参数量子群结构的形变量子振子超对称扩展，引入了pq形变超对称相干态，并证明其费米子与玻色子纠缠及坐标动量不确定关系具有非经典非最小特性。
We discuss the Bohmian mechanics by means of the deformed Schrödinger equation for position dependent mass, in the context of a $q$-algebra inspired by nonextensive statistics. A deduction of the Bohmian quantum formalism is performed by means of a deformed Fisher information functional, from which a deformed Cramér-Rao bound is derived. Lagrangian and Hamiltonian formulations, inherited by the $q$-algebra, are also developed. Then, we illustrate the results with a particle confined in an infinite square potential well. The preservation of the deformed Cramér-Rao bound for the stationary state…
A new version of the q-deformed supersymmetric quantum mechanics (q-SQM), which is inspired by the Tamm--Dankoff-type (TD-type) deformation of quantum harmonic oscillator, is constructed. The obtained algebra of q-SQM is similar to that in Spiridonov's approach. However, within our version of q-SQM, the ground state found explicitly in the special case of superpotential yiealding q-superoscillator turns out to be non-Gaussian and takes the form of special (TD-type) q-deformed Gaussian.
A new kind of deformed calculus (the D-deformed calculus) that takes place in fractional-dimensional spaces is presented. The D-deformed calculus is shown to be an appropriate tool for treating fractional-dimensional systems in a simple way and quite analogous to their corresponding one-dimensional partners. Two simple systems, the free particle and the harmonic oscillator in fractional- dimensional spaces are reconsidered into the framework of the D-deformed quantum mechanics. Confined states in a D-deformed quantum well are studied. D-deformed coherent states are also found.
Fractional supersymmetric quantum mechanics of order $λ$ is realized in terms of the generators of a generalized deformed oscillator algebra and a Z$_λ$-grading structure is imposed on the Fock space of the latter. This realization is shown to be fully reducible with the irreducible components providing $λ$ sets of minimally bosonized operators corresponding to both unbroken and broken cases. It also furnishes some examples of Z$_λ$-graded uniform topological symmetry of type (1, 1, ..., 1) with topological invariants generalizing the Witten index.
With a q-deformed quantum mechanical framework, features of the uncertainty relation and a novel formulation of the Schrödinger equation are considered.
Spectral and thermodynamic properties of supersymmetric quantum systems with self-adjoint deformed momentum
通过广义扩展动量算子构建严格自伴的变形动量，在非厄米超对称分解下解析求解线性与二次变形系统的谱，揭示隐藏对称性，并首次完整刻画其热力学性质，发现几何变形可调控量子热力学响应。
Supersymmetry applied to quantum mechanics has given new insights in various topics of theoretical physics like analytically solvable potentials, WKB approximation or KdV solitons. Duality plays a central role in many supersymmetric theories such as Yang-Mills theories or strings models. We investigate the possible existence of some duality within supersymmetric quantum mechanics.
The q-deformed coherent states for a quantum particle on a circle are introduced and their properties investigated.
We construct the exact position representation of a deformed quantum mechanics which exhibits an intrinsic maximum momentum and use it to study problems such as a particle in a box and scattering from a step potential, among others. In particular, we show that unlike usual quantum mechanics, the present deformed case delays the formation of bound states in a finite potential well. In the process we also highlight some limitations and pit-falls of low-momentum or perturbative treatments and thus resolve two puzzles occurring in the literature.
In order to realize supersymmetric quantum mechanics methods on a four dimensional classical phase-space, the complexified Clifford algebra of this space is extended by deforming it with the Moyal star-product in composing the components of Clifford forms. Two isospectral matrix Hamiltonians having a common bosonic part but different fermionic parts depending on four real-valued phase space functions are obtained. The Hamiltonians are doubly intertwined via matrix-valued functions which are divisors of zero in the resulting Moyal-Clifford algebra. Two illustrative examples corresponding to Jay…
Starting on the basis of $q$-symmetric oscillator algebra and on the associate $q$-calculus properties, we study a deformed quantum mechanics defined in the framework of the basic square-integrable wave functions space. In this context, we introduce a deformed Schroedinger equation, which satisfies the main quantum mechanics assumptions and admits, in the free case, plane wave functions that can be expressed in terms of the q-deformed exponential, originally introduced in the framework of the basic-hypergeometric functions.
We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translationally) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, by an infinite set of partial differential equations can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to $\hbar$-independent superpotentials and to a superpotential explicitly depending on $\…
We study the spectrum generating closed nonlinear superconformal algebra that describes $\mathcal{N}=2$ super-extensions of rationally deformed quantum harmonic oscillator and conformal mechanics models with coupling constant $g=m(m+1)$, $m\in {\mathbb N}$. It has a nature of a nonlinear finite $W$ superalgebra being generated by higher derivative integrals, and generally contains several different copies of either deformed superconformal $\mathfrak{osp}(2|2)$ algebra in the case of super-extended rationally deformed conformal mechanics models, or deformed super-Schrodinger algebra in the case…
Within the formulation of a q-deformed Quantum Mechanics a qualitative undercut of the q-deformed uncertainty relation from the Heisenberg uncertainty relation is revealed. When $q$ is some fixed value not equal to one, recovering of ordinary quantum mechanics and the corresponding recovering condition are discussed.
Starting on the basis of the non-commutative q-differential calculus, we introduce a generalized q-deformed Schrödinger equation. It can be viewed as the quantum stochastic counterpart of a generalized classical kinetic equation, which reproduces at the equilibrium the well-known q-deformed exponential stationary distribution. In this framework, q-deformed adjoint of an operator and q-hermitian operator properties occur in a natural way in order to satisfy the basic quantum mechanics assumptions.
An approach to study a generalization of the classical-quantum transition for general systems is proposed. In order to develop the idea, a deformation of the ladder operators algebra is proposed that contains a realization of the quantum group $SU(2)_q$ as a particular case. In this deformation Planck's constant becomes an operator whose eigenvalues approach $\hbar $ for small values of $n$ (the eigenvalue of the number operator), and zero for large values of $n$ (the system is classicalized).
Pöschl-Teller-driven solutions for quantum mechanical fluctuations are triggered off by single scalar field theories obtained through a systematic perturbative procedure for generating deformed defects. The analytical properties concerning the quantum fluctuations in one-dimension, zero-mode states, first- and second- excited states, and energy density profiles are all obtained from deformed topological and non-topological structures supported by real scalar fields. Results are firstly derived from an integrated $λϕ^4$ theory, with corresponding generalizations applied to starting $λχ^4$ and {…
We present the quantum and classical mechanics formalisms for a particle with position-dependent mass in the context of a deformed algebraic structure (named $κ$-algebra), motivated by the Kappa-statistics. From this structure we obtain deformed versions of the position and momentum operators, which allow to define a point canonical transformation that maps a particle with constant mass in a deformed space into a particle with position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews-Lakshmanan oscillator, …
A connection between fractional supersymmetric quantum mechanics and ordinary supersymmetric quantum mechanics is established in this Letter.
We revisit $T\bar T$ deformations of $d=2$ theories with fermions with a view toward the quantization. As a simple illustration, we compute the deformed Dirac bracket for a Majorana doublet and confirm the known eigenvalue flows perturbatively. We mostly consider those $T\bar T$ theories that can be reconstructed from string-like theories upon integrating out the worldsheet metric. After a quick overview of how this works when we add NSR-like or GS-like fermions, we obtain a known non-supersymmetric $T\bar T$ deformation of a $\cN=(1,1)$ theory from the latter, based on the Noether energy-mome…
We examine a deformed quantum mechanics in which the commutator between coordinates and momenta is a function of momenta. The Jacobi identity constraint on a two-parameter class of such modified commutation relations (MCR's) shows that they encode an intrinsic maximum momentum; a sub-class of which also imply a minimum position uncertainty. Maximum momentum causes the bound state spectrum of the one-dimensional harmonic oscillator to terminate at finite energy, whereby classical characteristics are observed for the studied cases. We then use a semi-classical analysis to discuss general concave…
We investigate the Weyl-Wigner-Gröenewold-Moyal, the Stratonovich and the Berezin group quantization schemes for the space-space noncommutative Heisenberg-Weyl group. We show that the $\star$-product for the deformed algebra of Weyl functions for the first scheme is different than that for the other two, even though their respective quantum mechanics' are equivalent as far as expectation values are concerned, provided that some additional criteria are imposed on the implementation of this process. We also show that it is the $\star$-product associated with the Stratonovich and the Berezin form…
The quantum oscillator and Kepler-Coulomb problems in $d$-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrödinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Ke…
We show that the formalism of supersymmetric quantum mechanics applied to the solvable elliptic function potentials $V(x) = mj(j+1){sn}^2(x,m)$ produces new exactly solvable one-dimensional periodic potentials.
We present a novel isospectral deformation chain in supersymmetric quantum mechanics. The chain is applied to the Coulomb potential.
Geometric Quantum Mechanics in a Symplectic Framework: Metric-Affine Extensions and Deformed Quantum Dynamics
本文基于项目ive希尔伯特空间的辛结构，提出量子力学的几何形式化，引入度量-仿射背景几何扩展，导致哈密顿流变形，并通过实例展示曲率和仿射结构对量子动力学的影响。
It is shown that the Hamilton equations in supersymmetric quantum mechanics can be presented in nonassociative form, where the Hamiltonian is decomposed into two nonassociative factors.
In this article, after introducing a kind of q-deformation in quantum mechanics, first, q-deformed form of Dirac equation in relativistic quantum mechanics is derived. Then three important scat erring problem in physics are studied. All results have satisfied what we had expected before. Furthermore, effects of all parameters in the problems on the reflection and transmission coefficients are calculated and shown graphically.
The Schrödinger equation is shown to be equivalent to a constrained Liouville equation under the assumption that phase space is extended to Grassmann algebra valued variables. For onedimensional systems, the underlying Hamiltonian dynamics has a N=2 supersymmetry. Potential applications to more realistic theories are briefly discussed.
This paper constitutes a review on N=2 fractional supersymmetric Quantum Mechanics of order k. The presentation is based on the introduction of a generalized Weyl-Heisenberg algebra W_k. It is shown how a general Hamiltonian can be associated with the algebra W_k. This general Hamiltonian covers various supersymmetrical versions of dynamical systems (Morse system, Poschl-Teller system, fractional supersymmetric oscillator of order k, etc.). The case of ordinary supersymmetric Quantum Mechanics corresponds to k=2. A connection between fractional supersymmetric Quantum Mechanics and ordinary sup…
We review the construction of minimally bosonized supersymmetric quantum mechanics and its relation to hidden supersymmetries in pure parabosonic (parafermionic) systems.
After recalling different formulations of the definition of supersymmetric quantum mechanics given in the literature, we discuss the relationships between them in order to provide an answer to the question raised in the title.
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