Position-dependent Mass Hamiltonian Research Articles

Quantum solvability of a nonlinear δ-type mass profile system: coupling constant quantization

In this paper, we discuss the quantum dynamics of a nonlinear system that admits temporally localized solutions at the classical level. We consider a general ordered position-dependent mass Hamiltonian in which the ordering parameters of the mass term are treated as arbitrary. The mass function here is singular at the origin. We observe that the quantum system admits bounded solutions but importantly the coupling parameter of the system gets quantized which has also been confirmed by the semiclassical study as well.

On the classical and quantum dynamics of a class of nonpolynomial oscillators

We consider two one dimensional nonlinear oscillators, namely (i) Higgs oscillator and (ii) a k-dependent nonpolynomial rational potential, where k is the constant curvature of a Riemannian manifold. Both the systems are of position dependent mass (PDM) form, , belonging to the quadratic Linard type nonlinear oscillators. They admit different kinds of motions at the classical level. While solving the quantum versions of the systems, we consider a generalized PDM Hamiltonian in which the ordering parameters of the mass term are treated as arbitrary. We observe that the quantum version of the Higgs oscillator is exactly solvable under appropriate restrictions of the ordering parameters, while the second nonlinear system is shown to be quasi exactly solvable using the Bethe ansatz method in which the arbitrariness of ordering parameters also plays an important role to obtain quasi-polynomial solutions. We extend the study to three dimensional generalizations of these nonlinear oscillators and obtain the exact solutions for the classical and quantum versions of the three dimensional Higgs oscillator. The three dimensional generalization of the quantum counterpart of the k-dependent nonpolynomial potential is found out to be quasi exactly solvable.

Quantum solvability of quadratic Liénard type nonlinear oscillators possessing maximal Lie point symmetries: An implication of arbitrariness of ordering parameters

In this paper, we investigate the quantum dynamics of underlying two one-dimensional quadratic Liénard type nonlinear oscillators which are classified under the category of maximal (eight parameter) Lie point symmetry group [Tiwari A K, Pandey S N, Senthilvelan M and Lakshmanan M 2013 J. Math. Phys. 54, 053 506]. Classically, both the systems were also shown to be linearizable as well as isochronic. In this work, we study the quantum dynamics of the nonlinear oscillators by considering a general ordered position dependent mass Hamiltonian. The ordering parameters of the mass term are treated to be arbitrary to start with. We observe that the quantum version of these nonlinear oscillators are exactly solvable provided that the ordering parameters of the mass term are subjected to certain constraints imposed on the arbitrariness of the ordering parameters. We obtain the eigenvalues and eigenfunctions associated with both the systems. We also consider briefly the quantum versions of other examples of quadratic Liénard oscillators which are classically linearizable.

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.

Isospectral Trigonometric Pöschl-Teller Potentials with Position Dependent Mass Generated by Supersymmetry

In this work a position dependent mass Hamiltonian with the same spectrum of the trigonometric Pöschl-Teller one was constructed by means of the underlying potential algebra. The corresponding wave functions are determined by using the factorization method. A new family of isospectral potentials are constructed by applying a Darboux transformation. An example is presented in order to illustrate the formalism.

Removal of ordering ambiguity for a class of position dependent mass quantum systems with an application to the quadratic Liénard type nonlinear oscillators

We consider the problem of removal of ordering ambiguity in position dependent mass quantum systems characterized by a generalized position dependent mass Hamiltonian which generalizes a number of Hermitian as well as non-Hermitian ordered forms of the Hamiltonian. We implement point canonical transformation method to map one-dimensional time-independent position dependent mass Schrödinger equation endowed with potentials onto constant mass counterparts which are considered to be exactly solvable. We observe that a class of mass functions and the corresponding potentials give rise to solutions that do not depend on any particular ordering, leading to the removal of ambiguity in it. In this case, it is imperative that the ordering is Hermitian. For non-Hermitian ordering, we show that the class of systems can also be exactly solvable and is also shown to be iso-spectral using suitable similarity transformations. We also discuss the normalization of the eigenfunctions obtained from both Hermitian and non-Hermitian orderings. We illustrate the technique with the quadratic Liénard type nonlinear oscillators, which admit position dependent mass Hamiltonians.

Exact Solutions of the Dirac Hamiltonian on the Sphere under Hyperbolic Magnetic Fields

Two-dimensional massless Dirac Hamiltonian under the influence of hyperbolic magnetic fields is mentioned in curved space. Using a spherical surface parameterization, the Dirac operator on the sphere is presented and the system is given as two supersymmetric partner Hamiltonians which coincides with the position dependent mass Hamiltonians. We introduce two ansatzes for the component of the vector potential to acquire effective solvable models, which are Rosen-Morse II potential and the model given Midya and Roy, whose bound states are JacobiX1type polynomials, and we adapt our work to these special models under some parameter restrictions. The energy spectrum and the eigenvectors are found for Rosen-Morse II potential. On the other hand, complete solutions are given for the second system. The vector and the effective potentials with their eigenvalues are sketched for each system.

Comparison Theorems for the Position-Dependent Mass Schrödinger Equation

The following comparison rules for the discrete spectrum of the position-dependent mass (PDM) Schrödinger equation are established. (i) If a constant mass m0 and a PDM m(x) are ordered everywhere, that is either, m0≤m(x) or m0≥m(x), then the corresponding eigenvalues of the constant-mass Hamiltonian and of the PDM Hamiltonian with the same potential and the BenDaniel-Duke ambiguity parameters are ordered. (ii) The corresponding eigenvalues of PDM Hamiltonians with the different sets of ambiguity parameters are ordered if ∇2(1/m(x)) has a definite sign. We prove these statements by using the Hellmann-Feynman theorem and offer examples of their application.

Nonsingular potentials from excited state factorization of a quantum system with position-dependent mass

The modified factorization technique of a quantum system characterized by a position-dependent mass Hamiltonian is presented. It has been shown that the singular superpotential defined in terms of a mass function and an excited state wavefunction of a given position-dependent mass Hamiltonian can be used to construct non-singular isospectral Hamiltonians. The method has been illustrated with the help of a few examples.

Exactly solvable position-dependent mass Hamiltonians related to non-compact semi-simple Lie groups

We suggest a generalized procedure to obtain exactly solvable position-dependent mass Hamiltonians in one dimension. The second-order Casimir invariant of the regular representation of a non-compact semi-simple Lie group G, the spectral properties of which are well known, is used to introduce exactly solvable Hamiltonians. A brief description of the procedure is presented and its application to quantum systems associated with SL(2, R) is detailed.

Quantum wave packet revival in two-dimensional circular quantum wells with position-dependent mass

We study quantum wave packet revivals on two-dimensional infinite circular quantum wells (CQWs) and circular quantum dots with position-dependent mass (PDM) envisaging a possible experimental realization. We consider CQWs with radially varying mass, addressing particularly the cases where M ( r ) ∝ r w with w = 1 , 2 , or −2. The two PDM Hamiltonians currently allowed by theory were analyzed and we were able to construct a strong theoretical argument favoring one of them.

First-Order Intertwining Operators with Position Dependent Mass and η-Weak-Pseudo-Hermiticity Generators

A Hermitian and an anti-Hermitian first-order intertwining operators are introduced and a class of η-weak-pseudo-Hermitian position-dependent mass (PDM) Hamiltonians are constructed. A corresponding reference-target η-weak-pseudo-Hermitian PDM—Hamiltonians’ map is suggested. Some η-weak-pseudo-Hermitian \(\mathcal{PT}\) -symmetric Scarf II and periodic-type models are used as illustrative examples. Energy-levels crossing and flown-away states phenomena are reported for the resulting Scarf II spectrum. Some of the corresponding η-weak-pseudo-Hermitian Scarf II- and periodic-type-isospectral models ( \(\mathcal{PT}\) -symmetric and non- \(\mathcal{PT}\) -symmetric) are given as products of the reference-target map.

Self-adjoint Hamiltonians with a mass jump: General matching conditions

The simplest position-dependent mass Hamiltonian in one dimension, where the mass has the form of a step function with a jump discontinuity at one point, is considered. The most general matching conditions at the jumping point for the solutions of the Schrodinger equation that provide a self-adjoint Hamiltonian are characterized.