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First-order intertwining operators and position-dependent mass Schrodinger equations in d dimensions

C. Quesne

Published Aug 29, 2005• 90 citations
OpenAlex

Abstract

The problem of d-dimensional Schrödinger equations with a position-dependent mass is analyzed in the framework of first-order intertwining operators. With the pair (H,H1) of intertwined Hamiltonians one can associate another pair of secondorder partial differential operators (R,R1), related to the same intertwining operator and such that H (resp. H1) commutes with R (resp. R1). This property is interpreted in superalgebraic terms in the context of supersymmetric quantum mechanics (SUSYQM). In the two-dimensional case, a solution to the resulting system of partial differential equations is obtained and used to build a physically-relevant model depicting a particle moving in a semi-infinite layer. Such a model is solved by employing either the commutativity of H with some second-order partial differential operator L and the resulting separability of the Schrödinger equation or that of H and R together with SUSYQM and shape-invariance techniques. The relation between both approaches is also studied.

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The problem of d-dimensional Schrödinger equations with a position-dependent mass is analyzed in the framework of first-order intertwining operators. With the pair (H,H1) of intertwined Hamiltonians one can associate another pair of secondorder partial differential operators (R,R1), related to the same intertwining operator and such that H (resp. H1) commutes with R (resp. R1). This property is interpreted in superalgebraic terms in the context of supersymmetric quantum mechanics (SUSYQM). In the two-dimensional case, a solution to the resulting system of partial differential equations is obtained and used to build a physically-relevant model depicting a particle moving in a semi-infinite layer. Such a model is solved by employing either the commutativity of H with some second-order partial differential operator L and the resulting separability of the Schrödinger equation or that of H and R together with SUSYQM and shape-invariance techniques. The relation between both approaches is also studied.
The interest in physical systems with position-dependent mass (PDM) is always topical. In this work we present a new way to construct kinetic energy operators (KEO’s) for particles endowed with a position varying mass within the Schrödinger equation. Taking recent developments of non-Hermitian quantum mechanics into account, we start with a non-Hermitian Hamiltonian and we recall that it is quasi-Hermitian and can be always associated to a Hermitian partner. The latter is then used along with the formalism of supersymmetric quantum mechanics (SUSY QM) to generate exactly solvable models on the basis of shape invariance condition. Some known and new concrete examples are considered as illustrations of the proposed scheme.
An exactly solvable position-dependent mass Schr\odinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical wavefunctions is stressed. Some new results for matrix elements are derived. This example emphasizes the interest of a quadratic algebra approach to position-dependent mass Schr\odinger equations.
We consider the application of the supersymmetric quantum-mechanical formalism to the Schr\"odinger equation describing a particle characterized by a position-dependent effective mass m(x).m(x).m(x). We show that any one-dimensional quantum system with effective mass has a supersymmetric partner system characterized by the same position dependence of the mass, but with a new potential function. The form of this supersymmetric partner potential V2(x){V}_{2}(x)V2​(x) depends on both the form of the original potential V1(x){V}_{1}(x)V1​(x) and the form of the mass x dependence. We also generalize the concept of shape invariance to the nonconstant mass scenario. As illustrative examples we provide, for a given form m(x)m(x)m(x) of the effective mass, shape-invariant potentials exhibiting (a) harmonic-oscillator-like spectra and (b) Morse-like spectra. In both cases the energy eigenvalues and eigenfunctions can be obtained in algebraic fashion.
Position-dependent e?ective mass systems are of great signi?cance due to the fact that these models have numerous applications in various areas of physics. The qualitative understanding of a complicated realistic system can be acquired by analyzing the exact solutions of a related simpli?ed model. However, the quantization of position-dependent e?ective mass systems and ?nding their solutions, involve some conceptual and mathematical di?culties of a fundamental nature. The factorization method provides us with a powerful tool for obtaining solutions and the underlying algebraic structure of the exactly solvable systems. The underlying algebra of a system has vast applications in di?erent areas of mathematics and physics, such as it plays an important role in the theory of coherent states. Coherent states are extremely useful in various areas such as quantum mechanics, quantum optics, quantum information and group theory. Inthisthesis, position-dependente?ectivemasssystemsarestudiedinthecontextoftheir quantization, ?nding the solutions, construction of the algebraic structure and associated coherent states. In the ?rst part of the thesis we mainly focus on quantizing and obtaining the exact solutions of the systems with spatially varying mass. The next part deals with the construction of the ladder operators and the inherent algebra of the pertaining systems. The associated coherent states and their properties are presented in the ?nal part of the thesis. Beside the traditional way of obtaining exact solutions by solving the Schrödinger equation there exists another elegant method to solve the systems algebraically by factorizing the corresponding Hamiltonian. This method is based on supersymmetric quantum mechanics and the integrability condition, commonly known as shape invariance. After quantizing the position-dependent e?ective mass system, this factorization technique is used to determine the energy spectrum and the corresponding wave functions. For the sake of completeness the \nmethod of solving a time-independent Schrödinger equation with spatially varying mass is also discussed. Considering a non-linear harmonic oscillator as an illustrative example, it is shown that both the above procedures produce the same results. The property of shape invariance enables us to obtain the ladder operators of the con?ning system. A general recipe for the construction of the ladder operators and inherent algebra for the position-dependent e?ective mass systems is presented. In order to exemplify the general formalism, a non-linear harmonic oscillator together with several other examples of the shape invariant systems with position-dependent e?ective mass is considered. Explicit expressions for the ladder operators and the associated algebra are presented. Usingtheladderoperatorsandtheunderlyingalgebra,thecoherentstatesforthepositiondependent e?ective mass systems are constructed and their properties are analyzed. In particular, we emphasize on various kinds of coherent states for a non-linear harmonic oscillator with spatially varying mass. By realizing SU(1,1) as the dynamic group of the system, the construction of Barut-Girardello coherent states is presented. In addition, an algebraic independent kind of coherent states, namely Gazeau-Klauder coherent states, are also constructed. The statistical properties of Barut-Girardello and Gazeau-Klauder coherent states are investigated by means of the Mandel parameter and the second order correlation function. Moreover, the temporal evolution of the Gazeau-Klauder coherent states is analysed by means of autocorrelation function. It is shown that these states mimic the phenomena of quantum revivals and fractional revivals during their time evolution
The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinetic energy operator (KEO) within the Schrodinger equation, in order to construct a new class of exactly solvable models with a position varying mass, presenting a harmonic-oscillator-like spectrum. To do so we utilize the formalism of supersymmetric quantum mechanics (SUSY QM) along with the shape invariance condition. Recent outcomes of non-Hermitian quantum mechanics are also taken into account.