Shape Invariant Potentials Research Articles

A unified scheme of central symmetric shape-invariant potentials

A unified scheme of central symmetric shape-invariant potentials

Supersymmetric Expansion Algorithm and Complete Analytical Solution for the Hulthén and Anharmonic Potentials

Abstract An algorithm for providing analytical solutions to Schrödinger’s equation with non-exactly solvable potentials is elaborated. It represents a symbiosis between the logarithmic expansion method and the techniques of the supersymmetric quantum mechanics as extended toward non shape invariant potentials. The complete solution to a given Hamiltonian H0 is obtained from the nodeless states of the Hamiltonian H0 and of a set of supersymmetric partners H1, H2, …, Hr. The nodeless states (dubbed “edge” states) are unique and in general can be ground or excited states. They are solved using the logarithmic expansion which yields an infinite system of coupled first order hierarchical differential equations, converted later into algebraic equations with recurrence relations which can be solved order by order. We formulate the aforementioned scheme, termed to as “Supersymmetric Expansion Algorithm” step by step and apply it to obtain for the first time the complete analytical solutions of the three dimensional Hulthén–, and the one-dimensional anharmonic oscillator potentials.

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

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 Ti, i=1,…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.

Anti-[formula omitted] transformations and complex non-Hermitian [formula omitted]-symmetric superpartners

We propose a new algebraic formalism for constructing complex non-Hermitian PT-symmetric superpartners by extending a conventional shape-invariant superpotential into the complex domain. The resulting potential is an unbroken super- and parity-time (PT)-symmetric shape-invariant potential with real energy eigenvalues, maintaining this property for all parameter values. In order to restore the probabilistic interpretation within a true quantum theory, a new inner product called the CPT-inner product is defined in PT-symmetric quantum mechanics, replacing the Dirac Hermitian inner product. In this work, we propose a new version of the inner product called the anti-PT (APT)-inner product, 〈A|B〉≡|A〉APT.|B〉, which replaces the previous versions without any additional considerations. This PT-supersymmetric quantum mechanics framework also allows for the unification of various areas of physics, including classical optics and quantum mechanics. To validate the theory, we present exact solutions for optical waveguides and the quantum tunneling probability, demonstrating excellent agreement with experimental data for the probability of crossing the potential barrier in the 3H(d,n)4He reaction.

Fractal Divergences of Generalized Jacobi Polynomials

The notion of entropy (including macro state entropy and information entropy) is used, among others, to define the fractal dimension. Rényi entropy constitutes the basis for the generalized correlation dimension of multifractals. A motivation for the study of the information measures of orthogonal polynomials is because these polynomials appear in the densities of many quantum mechanical systems with shape-invariant potentials (e.g., the harmonic oscillator and the hydrogenic systems). With the help of a sequence of some generalized Jacobi polynomials, we define a sequence of discrete probability distributions. We introduce fractal Kullback–Leibler divergence, fractal Tsallis divergence, and fractal Rényi divergence between every element of the sequence of probability distributions introduced above and the element of the equiprobability distribution corresponding to the same index. Practically, we obtain three sequences of fractal divergences and show that the first two are convergent and the last is divergent.

Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory

Darboux developed an algebraic mechanism to construct an infinite chain of second order differential equations as well as their solutions. After a surprisingly long time, Darboux's results had important features in the analytic context, for instance in quantum mechanics where it provides a convenient framework for Supersymmetric Quantum Mechanics. Today, there are a lot of papers regarding the use of Darboux transformations in various contexts, not only in mathematical physics. In this paper, we develop a generalization of the Darboux transformations for tensor product constructions on linear differential equations or systems. Moreover, we provide explicit Darboux transformations for $\sym^2 (\mathrm{SL}(2,\mathbb{C}))$ systems and, as a consequence, also for $\mathfrak{so}(3, C_K)$ systems, to construct an infinite chain of integrable (in Galois sense) linear differential systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials.

SWKB quantization condition for conditionally exactly solvable systems and the residual corrections

The SWKB quantization condition is an exact quantization condition for the conventional shape-invariant potentials. On the other hand, this condition equation does not hold for other known solvable systems. The origin of the (non-)exactness is understood in the context of the quantum Hamilton–Jacobi formalism. First we confirm the statement and show inexplicit properties numerically for the case of the conditionally exactly solvable systems by Junker and Roy. The SWKB condition clearly breaks for this case, but the condition equation is restored within a certain degree of accuracy. We propose a novel approach to evaluate the residual by perturbation, intending to explore what the correction terms for the SWKB condition equation look like.

A note on shape invariant potentials for discretized Hamiltonians

Using the method of the “exact discretization” of the Schrödinger equation, we propose a particular discretized version of the [Formula: see text] Supersymmetric Quantum Mechanics (SQM). After defining the corresponding shape invariance condition, we show that the energy spectra and wave functions for discretized Quantum Mechanical systems can be found using the technique of [Formula: see text] Supersymmetric Quantum Mechanics exactly the same way as it is done for their continuous counterparts. As a demonstration of the present method, we find the energy spectrum for a discretized Coulomb potential and its ground state wave function.

Shape Invariant Potentials in Supersymmetric Quantum Cosmology

In this brief review, we comment on the concept of shape invariant potentials, which is an essential feature in many settings of N=2 supersymmetric quantum mechanics. To motivate its application within supersymmetric quantum cosmology, we present a case study to illustrate the value of this promising tool. Concretely, we take a spatially flat FRW model in the presence of a single scalar field, minimally coupled to gravity. Then, we extract the associated Schrödinger–Wheeler–DeWitt equation, allowing for a particular scope of factor ordering. Subsequently, we compute the corresponding supersymmetric partner Hamiltonians, H1 and H2. Moreover, we point out how the shape invariance property can be employed to bring a relation among several factor orderings choices for our Schrödinger–Wheeler–DeWitt equation. The ground state is retrieved, and the excited states easily written. Finally, the Hamiltonians, H1 and H2, are explicitly presented within a N=2 supersymmetric quantum mechanics framework.

Infinite families of shape invariant potentials with [formula omitted] parameters subject to translation

We find new families of shape invariant potentials depending on n≥1 parameters subject to translation by the inclusion of non-trivial invariants. New dependencies of the spectra are found, and it opens the door to the engineering of physical quantities in a novel way. A number of examples are explicitly constructed.

One parameter family of rationally extended isospectral potentials

We start from a given one dimensional rationally extended shape invariant potential associated with Xm exceptional orthogonal polynomials and using the idea of supersymmetry in quantum mechanics, we obtain one continuous parameter (λ) family of rationally extended strictly isospectral potentials. We illustrate this construction by considering three well known rationally extended potentials, two with pure discrete spectrum (the extended radial oscillator and the extended Scarf-I) and one with both the discrete and the continuous spectrum (the extended generalized Pöschl–Teller) and explicitly construct the corresponding one continuous parameter family of rationally extended strictly isospectral potentials. Further, in the special case of λ=0 and −1, we obtain two new exactly solvable rationally extended potentials, namely the rationally extended Pursey and the rationally extended Abraham–Moses potentials respectively. We illustrate the whole procedure by discussing in detail the particular case of the X1 rationally extended one parameter family of potentials including the corresponding Pursey and the Abraham Moses potentials.

Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ

We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translational) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, which, 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 ℏ-independent superpotentials and to a superpotential explicitly depending on ℏ.

Electron in bilayer graphene with magnetic fields leading to shape invariant potentials

The quantum behavior of electrons in bilayer graphene with applied magnetic fields is addressed. By using second-order supersymmetric quantum mechanics the problem is transformed into two intertwined one dimensional stationary Schrödinger equations whose potentials are required to be shape invariant. Analytical solutions for the energy bound states are obtained for several magnetic fields. The associated spectrum is analyzed, and the probability and current densities are determined.

Exactness of SWKB for shape invariant potentials

The supersymmetry-based semiclassical method (SWKB) is known to produce exact spectra for conventional shape invariant potentials. In this paper we prove that this exactness follows from their additive shape invariance.

Relativistic potentials with rational extensions

In this paper, we construct isospectral Hamiltonians without shape invariant potentials for the relativistic quantum mechanical potentials such as the Dirac Oscillator and Hydrogen-like atom.

Study of Bohr–Mottelson with minimal length effect for Q-deformed modified Eckart potential and Bohr–Mottelson with Q-deformed quantum for three-dimensional harmonic oscillator potential

Bohr–Mottelson Hamiltonian on the γ-rigid regime for Q-deformed modified Eckart and three-dimensional harmonic oscillator potentials in the β-collective shape variable was investigated in the presence of minimal length formalism and Q-deformed of the radial momentum part. By introducing new wave function and using the Q-deformed potential concept in Bohr–Mottelson Hamiltonian in the minimal length formalism, the un-normalized wave function and energy spectra equation were obtained by using the hypergeometric method. Meanwhile, the Bohr–Mottelson Hamiltonian in the presence of the quadratic spatial deformation to the momentum in collective shape variable was investigated using transformation of a new variable such as the Schrodinger-like equation with shape invariant potential. The energy equation and un-normalized wave function were obtained using the hypergeometric method. The results showed that the Bohr–Mottelson equations with different energy potentials and different deformation forms in the radial momentum reduced to the similar Schrodinger-like equation with the modified Poschl–Teller potential.

Construction of solvable non-central potential using vector superpotential: a new approach

We introduce here vector superpotential which is useful to find general potential form using supersymmetric quantum mechanics (SUSY QM) approach. Using the vector superpotential, we reconstruct different solved central potentials and non-central shape variant or shape invariant potentials. To construct and reconstruct the central potentials and non-central potentials, appropriate choice of radial part of the superpotential as well as angular and azimuthal parts of the superpotential is required. Our main aim is to construct angular and azimuthal parts of potential directly from the appropriate choice of vector superpotential. We have reconstructed few potentials which are solved by different researchers using different methods of solution and also constructed new non-central potentials.

Exact Solutions of Schrodinger Equation in Cylindrical Coordinates for Double Ring-Shaped Coulomb Oscillator Potential Using SUSY QM Method

Schrodinger equation for a doubled ring-shaped coulomb oscillator potential is investigated using supersymmetric quantum mechanics approach. The three dimensional Schrodinger equation for a doubled ring-shaped coulomb oscillator potential in cylindrical coordinates is separated into three parts that contains one dimensional Schrodinger type equation which are solved using supersymmetric operator and the idea of shape invariant potential. The energy spectrum and the total wave functions are obtained.

Exact results for the infinite supersymmetric extensions of the infinite square well

One-dimensional potentials defined by (for integer S) arise in the repeated supersymmetrization of the infinite square well, here defined over the region (0, a). We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closed-form expressions for the normalised solutions, , for all V(S)(x) in terms of well-known special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shape-invariant potentials, the trigonometric Pöschl–Teller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behaviour of the corresponding momentum-space wave functions for large and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.

The supersymmetric WKB formalism is not exact for all additive shape invariant potentials

Following the verification of the conjecture made by Comtet et al that the supersymmetry-inspired semiclassical method known as SWKB is exact for the conventional additive shape invariant potentials, it was widely believed that SWKB yields exact results for all additive shape invariant potentials. In this paper we present a concrete example of an additive shape invariant potential for which the SWKB method fails to produce exact results.