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13 pages, 333 KB

Open AccessArticle

The Second-Order Approximation of Superpotentials Based on SUSYQM

by Yao Liu, Yin Yin, Wenxin Qiu, Wei Cheng, Huan Lu and Guang Luo

Symmetry 2025, 17(4), 493; https://doi.org/10.3390/sym17040493 - 25 Mar 2025

Viewed by 234

This paper is based on the shape invariance of the solvable superpotentials and uses the series expansion method to study the approximate expansion forms of these superpotentials. Firstly, this paper examines the differential equations satisfied by the first-order approximations of the superpotentials. Through an example, namely Rosen–Morse (trigonometric) superpotentials, the specific forms of these first-order approximations are analyzed. Based on these simple first-order approximations, this paper then delves into the ground-state wave functions of the superpotential. Secondly, this paper derives the differential equations satisfied by the second-order approximations with the first-order approximations. Using the harmonic oscillator superpotentials as an example, similarly, non-unique forms for the second-order approximations are obtained. By selecting simpler forms for the first- and second-order approximations, the authors further investigate the ground-state wave functions of the superpotential with the second-order approximation. Thirdly, the authors discuss the Hamiltonians of the potential with the first- and second-order approximations, concluding that the additional term originates from the corrections to the superpotential. Finally, conclusions and prospects are provided. Full article

(This article belongs to the Section Physics)

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19 pages, 386 KB

Open AccessArticle

Strained Graphene as Pristine Graphene with a Deformed Momentum Operator

by David Valenzuela, Alfredo Raya and Juan D. García-Muñoz

Condens. Matter 2025, 10(1), 10; https://doi.org/10.3390/condmat10010010 - 7 Feb 2025

Viewed by 808

Abstract

We explore the equivalence between the low-energy dynamics of strained graphene and a quantum mechanical framework for the 2D Dirac equation in flat space with a deformed momentum operator. By considering some common forms of the anisotropic Fermi velocity tensor emerging from the elasticity theory, we associate such tensor forms with a deformation of the momentum operator. We first explore the bound states of charge carriers in a background uniform magnetic field in this framework and quantify the impact of strain in the energy spectrum. Then, we use a quadrature algebra formula as a mathematical tool to analyze the impact of the deformation attached to the momentum operator and identify physical consequences of such deformation in terms of energy modifications due to the applied strain. Full article

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25 pages, 398 KB

Open AccessArticle

Unified Supersymmetric Description of Shape-Invariant Potentials within and beyond the Natanzon Class

by Tibor Soltész, Levente Ferenc Pethő and Géza Lévai

Symmetry 2024, 16(2), 174; https://doi.org/10.3390/sym16020174 - 1 Feb 2024

Cited by 4 | Viewed by 1898

Abstract

The transformations of supersymmetric quantum mechanics are discussed within a formalism that employs a six-parameter function, from which the superpotential and the supersymmetric partner potentials

V_{-} (x)

and

V_{+} (x)

are constructed in a general form. By [...] Read more.

The transformations of supersymmetric quantum mechanics are discussed within a formalism that employs a six-parameter function, from which the superpotential and the supersymmetric partner potentials

V_{-} (x)

and

V_{+} (x)

are constructed in a general form. By specific choice of the parameters,

V_{-} (x)

and

V_{+} (x)

are matched with the general form of PI class potentials and their rationally extended versions. The choice of the parameters also determines which of the four possible SUSY transformations

T_{i}

i = 1, \dots 4

is in effect. After this general discussion, the formulae are specified to the three members of this potential class, the Scarf I, Scarf II and generalized Pöschl–Teller potentials. Due to the different domains of definition and their consequences on the boundary conditions, the results turn out to be rather diverse for the three potentials, while the mathematical formalism and the network of the potentials interconnected by the SUSYQM transformations still remains common to a large extent. The general framework allows a unified and consistent interpretation of earlier isolated findings. It also helps to connect the results to further potential classes and to place them into a more general context within the zoo of exactly solvable potentials. Full article

(This article belongs to the Section Physics)

18 pages, 874 KB

Open AccessArticle

A New Solvable Generalized Trigonometric Tangent Potential Based on SUSYQM

by Lulin Xiong, Xin Tan, Shikun Zhong, Wei Cheng and Guang Luo

Symmetry 2022, 14(8), 1593; https://doi.org/10.3390/sym14081593 - 3 Aug 2022

Cited by 2 | Viewed by 1931

Abstract

Supersymmetric quantum mechanics has wide applications in physics. However, there are few potentials that can be solved exactly by supersymmetric quantum mechanics methods, so it is undoubtedly of great significance to find more potentials that can be solved exactly. This paper studies the [...] Read more.

A \tan n p x + B \tan m p x

. We will elaborate on this new potential in the following aspects. Firstly, the shape invariant relation of partner potential is generated by the generalized trigonometric tangent superpotential. We find three shape invariance forms that satisfy the additive condition. Secondly, the eigenvalues and the eigenwave functions of the potential are studied separately in these three cases. Thirdly, the potential algebra of such a superpotential is discussed, and the discussions are explored from two aspects: one parameter’s and two parameters’ potential algebra. Through the potential algebra, the eigenvalue spectrums are given separately which are consistent with those mentioned earlier. Finally, we summarize the paper and give an outlook on the two-parameter shape-invariant potential. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

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16 pages, 726 KB

Open AccessArticle

Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

by Sara Cruz y Cruz, Alejandro Romero-Osnaya and Oscar Rosas-Ortiz

Symmetry 2021, 13(9), 1583; https://doi.org/10.3390/sym13091583 - 27 Aug 2021

Cited by 6 | Viewed by 2592

Abstract

The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux–Crum transformations. We apply the finite-difference algorithm for higher-order supersymmetric quantum mechanics to obtain complex-valued refractive indices admitting all-real eigenvalues in their point spectrum. The new refractive indices are such that their imaginary part gives zero if it is integrated over the entire domain of definition. This property, called condition of zero total area, ensures the conservation of optical power so the refractive index shows balanced gain and loss. Consequently, the complex-valued refractive indices reported in this work include but are not limited to the parity-time invariant case. Full article

(This article belongs to the Section Physics)

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22 pages, 507 KB

Open AccessReview

Supersymmetric Quantum Mechanics and Solvable Models

by Jonathan Bougie, Asim Gangopadhyaya, Jeffry Mallow and Constantin Rasinariu

Symmetry 2012, 4(3), 452-473; https://doi.org/10.3390/sym4030452 - 16 Aug 2012

Cited by 35 | Viewed by 8347

Abstract

We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of ħ-independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on ħ. Full article

(This article belongs to the Special Issue Supersymmetry)

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