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arXiv:1601.06735 · Journal of Physics A Mathematical and Theoretical

Position-dependent mass quantum Hamiltonians: general approach and duality

M. A. Rego-Monteiro, Ligia M. C. S. Rodrigues, Evaldo M. F. Curado

Published Feb 9, 2016• 18 citations
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Abstract

Abstract We analyze a general family of position-dependent mass (PDM) quantum Hamiltonians which are not self-adjoint and include, as particular cases, some Hamiltonians obtained in phenomenological approaches to condensed matter physics. We build a general family of self-adjoint Hamiltonians which are quantum mechanically equivalent to the non-self-adjoint proposed ones. Inspired by the probability density of the problem, we construct an ansatz for the solutions of the family of self-adjoint Hamiltonians. We use this ansatz to map the solutions of the time independent Schrödinger equations generated by the non-self-adjoint Hamiltonians into the Hilbert space of the solutions of the respective dual self-adjoint Hamiltonians. This mapping depends on both the PDM and on a function of position satisfying a condition that assures the existence of a consistent continuity equation. We identify the non-self-adjoint Hamiltonians here studied with a very general family of Hamiltonians proposed in a seminal article of Harrison (1961 Phys. Rev. 123 85 ) to describe varying band structures in different types of metals. Therefore, we have self-adjoint Hamiltonians that correspond to the non-self-adjoint ones found in Harrison’s article.

DOI
10.1088/1751-8113/49/12/125203

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arXiv:1710.02135•Journal of Physics A Mathematical and Theoretical•Sep 22, 2017•22 citations
Abstract We analyze a general family of position-dependent mass (PDM) quantum Hamiltonians which are not self-adjoint and include, as particular cases, some Hamiltonians obtained in phenomenological approaches to condensed matter physics. We build a general family of self-adjoint Hamiltonians which are quantum mechanically equivalent to the non-self-adjoint proposed ones. Inspired by the probability density of the problem, we construct an ansatz for the solutions of the family of self-adjoint Hamiltonians. We use this ansatz to map the solutions of the time independent Schrödinger equations generated by the non-self-adjoint Hamiltonians into the Hilbert space of the solutions of the respective dual self-adjoint Hamiltonians. This mapping depends on both the PDM and on a function of position satisfying a condition that assures the existence of a consistent continuity equation. We identify the non-self-adjoint Hamiltonians here studied with a very general family of Hamiltonians proposed in a seminal article of Harrison (1961 Phys. Rev. 123 85 ) to describe varying band structures in different types of metals. Therefore, we have self-adjoint Hamiltonians that correspond to the non-self-adjoint ones found in Harrison’s article.
Position-dependent mass (PDM) systems have diverse applications in physics, attracting significant scientific interest in the last decades. Classical PDM systems pose analytical challenges with their nonlinear equations of motion, while the quantum case is complicated because the kinetic operator is non-Hermitian. To address these challenges, this study explores Hamiltonian factorization and canonical transformations to investigate classical and quantum PDM systems, applying it to a Mathews-Lakshmanan-type oscillator with m⁢(x)=1/[1+(λ⁢x)2]. Classically, the phase space is examined, revealing increasingly pronounced deformities in the trajectories as energy and λ values increase. In the quantum realm, the solution to the ambiguous ordering problem for the PDM oscillator is presented, accompanied by an analysis of wave functions and probability densities. Further, the tunneling probabilities are analyzed. As λ increases, findings indicate that the tunneling probabilities of the PDM system decrease fast for higher excited states, offering novel insights into its behavior.
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.
We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems - the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone\'s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the time-dependent Schrödinger equation.
A linear one-dimensional Schr\"odinger equation, defined by means of a non-Hermitian Hamiltonian characterized by position-dependent masses, was proposed lately. Herein we present an exact classical field theory for this equation, showing the need for an extra field $\ensuremath{\Phi}(x,t)$, in addition to the usual one, $\ensuremath{\Psi}(x,t)$, similar to what was done recently in the analysis of a class of nonlinear quantum equations. These generalizations of the Schr\"odinger equation depend on an index qqq, in such a way that the standard case is recovered in the limit $q\ensuremath{\rightarrow}1$. Particularly, the field $\ensuremath{\Phi}(x,t)$ becomes ${\ensuremath{\Psi}}^{*}(x,t)$ only when $q\ensuremath{\rightarrow}1$ and satisfies a similar Schr\"odinger equation for the Hermitian conjugate of the Hamiltonian operator. In terms of these two fields one may define a probability density following a standard continuity equation, leading to the preservation of probability in Cartesian space. Simple applications are performed by solving the equations for the two fields.
Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA L{\mathcal{L}}L?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA H{\mathcal{H}}H?> H , the amended Hamiltonian H ≠ H † can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical “user-unfriendliness” of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.
Abstract The quantization of systems with a position dependent mass (PDM) is studied. We present a method that starts with the study of the existence of Killing vector fields for the PDM geodesic motion (Lagrangian with a PDM kinetic term but without any potential) and the construction of the associated Noether momenta. Then the method considers, as the appropriate Hilbert space, the space of functions that are square integrable with respect to a measure related with the PDM and, after that, it establishes the quantization, not of the canonical momenta p , but of the Noether momenta P instead. The quantum Hamiltonian, that depends on the Noether momenta, is obtained as an Hermitian operator defined on the PDM Hilbert space. In the second part several systems with position-dependent mass, most of them related with nonlinear oscillators, are quantized by making use of the method proposed in the first part.