Point Canonical Transformation Research Articles

Rational extensions of an oscillator-shaped quantum well potential in a position-dependent mass background

We show that a recently proposed oscillator-shaped quantum well model associated with a position-dependent mass can be solved by applying a point canonical transformation to the constant-mass Schrödinger equation for the Scarf I potential. On using the known rational extension of the latter connected with X 1-Jacobi exceptional orthogonal polynomials, we build a rationally-extended position-dependent mass model with the same spectrum as the starting one. Some more involved position-dependent mass models associated with X 2-Jacobi exceptional orthogonal polynomials are also considered.

Semi-infinite Quantum Wells In a Position-Dependent Mass Background

Using a point canonical transformation starting from the constant-mass Schrödinger equation for the Morse potential, it is shown that a semi-infinite quantum well model with a non-rectangular profile associated with a position-dependent mass that becomes infinite for some negative value of the position, while going to a constant for a large positive value of the latter can be easily derived. In addition, another type of semi-infinite quantum well associated with the same position-dependent mass is constructed and solved by starting from the Rosen-Morse II potential instead of the Morse one.

Exact solution and coherent states of an asymmetric oscillator with position-dependent mass

We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 62, 092101 (2021)] in the classical and quantum formalisms by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space (x, p) to a deformed one (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.

So(2, 1) algebra, local Fermi velocity, and position-dependent mass Dirac equation

We investigate the (1 + 1)-dimensional position-dependent mass Dirac equation within the confines of so(2, 1) potential algebra by utilizing the character of a spatial varying Fermi velocity. We examine the combined effects of the two when the Dirac equation is equipped with an external pseudoscalar potential. Solutions of the three cases induced by so(2, 1) are explored by profitably making use of a point canonical transformation.

Generalized semiconfined harmonic oscillator model with a position-dependent effective mass

By using a point canonical transformation starting from the constant-mass Schr\"odinger equation for the isotonic potential, it is shown that a semiconfined harmonic oscillator model with a position-dependent mass in the BenDaniel-Duke setting and the same spectrum as the standard harmonic oscillator can be easily constructed and extended to a semiconfined shifted harmonic oscillator, which could result from the presence of a uniform gravitational field. A further generalization is proposed by considering a $m$-dependent position-dependent mass for $0<m<2$ and deriving the associated semiconfined potential. This results in a family of position-dependent mass and potential pairs, to which the original pair belongs as it corresponds to $m=1$. Finally, the potential that would result from a general von Roos kinetic energy operator is presented and the examples of the Zhu-Kroemer and Mustafa-Mazharimousavi settings are briefly discussed.

Ladder operators and coherent states for the Rosen–Morse system and its rational extensions

Ladder operators for the hyperbolic Rosen–Morse (RMII) potential are realized using the shape invariance property appearing, in particular, using supersymmetric quantum mechanics. The extension of the ladder operators to a specific class of rational extensions of the RMII potential is presented and discussed. Coherent states are then constructed as almost eigenstates of the lowering operators. Some properties are analyzed and compared. The ladder operators and coherent states constructions presented are extended to the case of the trigonometric Rosen–Morse (RMI) potential using a point canonical transformation.

On the position-dependent mass Schrödinger equation for Mie-type potentials

The exactly solvable Position Dependent Mass Schrödinger Equation (PDMSE) for Mie-type potentials is presented. To that, by means of a point canonical transformation the exactly solvable constant mass Schrödinger equation is transformed into a PDMSE. The mapping between both Schrödinger equations lets obtain the energy spectra and wave functions for the potential under study. This happens for any selection of the O von Roos ambiguity parameters involved in the kinetic energy operator. The exactly solvable multiparameter exponential-type potential for the constant mass Schrödinger equation constitutes the reference problem allowing to solve the PDMSE for Mie potentials and mass functions of the form given by m(x) = skx s-1/(xs + 1))2. Thereby, as a useful application of our proposal, the particular Lennard-Jones potential is presented as an example of Mie potential by considering the mass distribution m(x) = 6kx 5/(x 6 + 1))2. The proposed method is general and can be straightforwardly applied to the solution of the PDMSE for other potential models and/or with different position-dependent mass distributions.

N-dimensional PDM-damped harmonic oscillators: linearizability, and exact solvability

We consider position-dependent mass (PDM) Lagrangians/Hamiltonians in their standard textbook form, where the long-standing gain-loss balance between the kinetic and potential energies is kept intact to allow conservation of total energy (i.e., , , and ). Under such standard settings, we discuss and report on n-dimensional PDM damped harmonic oscillators (DHO). We use some n-dimensional point canonical transformation to facilitate the linearizability of their n-PDM dynamical equations into some n-linear DHOs’ dynamical equations for constant mass setting. Consequently, the well know exact solutions for the linear DHOs are mapped, with ease, onto the exact solutions for PDM DHOs. A set of one-dimensional and a set of n-dimensional PDM-DHO illustrative examples are reported along with their phase-space trajectories.

Path integral formulation for position–time-dependent mass systems

The problem of position-time dependent mass with general time-dependent potential is treated in the framework of the phase space path integral approach. By means of an appropriate point canonical transformation, the problem is converted into that relative to a constant mass with stationary potential. Particular examples are also considered.

Revisiting an isospectral extension of the Morse potential

A very simple method is devised to derive a (strictly) isospectral extension of the Morse potential. Furthermore, point canonical transformations are used to transform the latter into quasi-exactly solvable extensions of the radial oscillator and the Coulomb potentials.

κ-Deformed quantum and classical mechanics for a system with position-dependent effective mass

We present the quantum and classical mechanics formalisms for a particle with a 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 us to define a point canonical transformation that maps a particle with a constant mass in a deformed space into a particle with a 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, exhibiting uncertainty relations depending on the deformation.

Generalized Extended Momentum Operator

We study and generalize the momentum operator satisfying the extended uncertainty principle relation (EUP). This generalized extended momentum operator (GEMO) consists of an arbitrary auxiliary function of position operator, , in such a combination that not only GEMO satisfies the EUP relation but also it is Hermitian. Next, we apply the GEMO to construct the generalized one-dimensional Schrödinger equation. Upon using the so called point canonical transformation (PCT), we transform the generalized Schrödinger equation from x-space to z-space where in terms of the transformed coordinate, z, it is of the standard form of the Schrödinger equation. In continuation, we study two illustrative examples and solve the corresponding equations analytically to find the energy spectrum.

Introducing supersymmetric quantum mechanics via point canonical transformations

We propose an alternative way to introduce supersymmetric quantum mechanics to physics students using a point canonical coordinate transformation. We provide two point canonical transformations that allow us to obtain new supersymmetric partner Hamiltonians with iso-spectral and non-iso-spectral energy eigenvalues starting from a known solvable Hamiltonian. We illustrate this new approach by applying it to the harmonic oscillator potential.

PDM creation and annihilation operators of the harmonic oscillators and the emergence of an alternative PDM-Hamiltonian

The exact solvability and impressive pedagogical implementation of the harmonic oscillator's creation and annihilation operators make it a problem of great physical relevance and the most fundamental one in quantum mechanics. So would be the position-dependent mass (PDM) oscillator for the PDM quantum mechanics. We, hereby, construct the PDM creation and annihilation operators for the PDM oscillator via two different approaches. First, via von Roos PDM Hamiltonian and show that the commutation relation between the PDM creation Aˆ+ and annihilation Aˆ operators, [Aˆ,Aˆ+]=1⇔AˆAˆ+−1/2=Aˆ+Aˆ+1/2, not only offers a unique PDM-Hamiltonian Hˆ1 but also suggests a PDM-deformation in the coordinate system. Next, we use a PDM point canonical transformation of the textbook constant mass harmonic oscillator analog and obtain yet another set of PDM creation Bˆ+ and annihilation Bˆ operators, hence an “apparently new” PDM-Hamiltonian Hˆ2 is obtained. The “new” PDM-Hamiltonian Hˆ2 turned out to be not only correlated with Hˆ1 but also represents an alternative and most simplistic user-friendly PDM-Hamiltonian, Hˆ=(pˆ/2m(x))2+V(x); pˆ=−iħ∂x, that has never been reported before.

Inter-relations between additive shape invariant superpotentials

All known additive shape invariant superpotentials in nonrelativistic quantum mechanics belong to one of two categories: superpotentials that do not explicitly depend on ħ, and their ħ-dependent extensions. The former group themselves into two disjoint classes, depending on whether the corresponding Schrödinger equation can be reduced to a hypergeometric equation (type-I) or a confluent hypergeometric equation (type-II). All the superpotentials within each class are connected via point canonical transformations. Previous work [19] showed that type-I superpotentials produce type-II via limiting procedures. In this paper we develop a method to generate a type I superpotential from type II, thus providing a pathway to interconnect all known additive shape invariant superpotentials.

Position-dependent mass momentum operator and minimal coupling: point canonical transformation and isospectrality

The classical and quantum-mechanical correspondence for constant mass settings is used, along with some point canonical transformation, to find the position-dependent mass (PDM) classical and quantum Hamiltonians. The comparison between the resulting quantum PDM-Hamiltonian and the von Roos PDM-Hamiltonian implied that the ordering ambiguity parameters of von Roos are strictly determined. Eliminating, in effect, the ordering ambiguity associated with the von Roos PDM-Hamiltonian. This, consequently, played a vital role in the construction and identification of the PDM-momentum operator. The same recipe is followed to identify the form of the minimal coupling of electromagnetic interactions for the classical and quantum PDM-Hamiltonians. It turned out that whilst the minimal coupling may very well inherit the usual form in classical mechanics (i.e., $p_{j} (\vec{x}) \rightarrow p_{j}(\vec{x}) -e A_{j}(\vec{x})$ , where $p_{j}(\vec{x})$ is the j-th component of the classical PDM-canonical-momentum), it admits a necessarily different and vital form in quantum mechanics (i.e., $\hat{p}_{j}(\vec{x})/\sqrt{m(\vec{x})} \rightarrow (\hat{p}_{j}(\vec{x})$ $ -e A_{j}(\vec{x}))/\sqrt{m(\vec{x})}$ , where $ \hat{p}_{j}(\vec{x})$ is the j-th component of the quantum PDM-momentum operator). Under our point transformation settings, only one of the two commonly used vector potentials (i.e., $\vec{A} (\vec{x}) \sim (-x_{2}, x_{1}, 0)$ ) is found eligible and is considered for our Illustrative examples.

RATIONALLY EXTENDED SHAPE INVARIANT POTENTIALS IN ARBITRARY D DIMENSIONS ASSOCIATED WITH EXCEPTIONAL Xm POLYNOMIALS

Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state solutions of these exactly solvable potentials can be written in terms of &lt;em&gt;X&lt;sub&gt;m&lt;/sub&gt;&lt;/em&gt; Laguerre or &lt;em&gt;X&lt;sub&gt;m&lt;/sub&gt;&lt;/em&gt; Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.

Quantum oscillator and Kepler–Coulomb problems in curved spaces: Deformed shape invariance, point canonical transformations, and rational extensions

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 Kepler-Coulomb potentials in curved space. The oscillator on the sphere and the Kepler-Coulomb potential in a hyperbolic space are studied in detail and their extensions are proved to be consistent with already known ones in Euclidean space. The partnership between nonextended and extended potentials is interpreted in a deformed supersymmetric framework. Those extended potentials that are isospectral to some nonextended ones are shown to display deformed shape invariance, which in the Kepler-Coulomb case is enlarged by also translating the degree of the polynomial arising in the rational part denominator.

The role of shape invariance potentials in the relativistic quantum mechanics

The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the solutions of the Dirac equation with spherical symmetry electromagnetic potentials. We want to show that some of the non-relativistic solvable potentials with shape-invariant symmetry can be related to the radial Dirac equation. Using this method, the idea of supersymmetry and shape invariance can be expanded to the relativistic quantum mechanics. The spinor wave functions for some of the obtained four-component electromagnetic potential are given in terms of special functions such as Jacobi, generalized Laguerre and Hermite polynomials. The relativistic bound-states spectrum for each case is also calculated in terms of the bound-states spectrum of the solvable potentials.

Effective mass Schrödinger equation with Thomas-Fermi potential

The exactly-solvable position-dependent mass Schrodinger equation (PDMSE) for the Thomas-Fermi potential is presented. The PDMSE is transformed into a standard Schrodinger equation (SSE) with constant mass by means of a point canonical transformation scheme. By proposing an exponential type potential of the SSE it is possible to determine a PDMSE with the Thomas-Fermi potential. The resulting PDMSE is carried to the Sturm- Liouville form and the corresponding theory is developed for the particular resulting problem in order to obtain closed solutions in terms of Bessel functions of the first kind. Beyond the case considered in this work, the approach is general and can be useful in the study of electronic properties of non-uniform materials in which the carrier effective mass depends on the position as well as in the search of new solvable potentials suitable for quantum systems.