Position-dependent mass quantum Hamiltonians: general approach and duality

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2016-03, Vol.49 (12), p.125203
• Main Authors: Rego-Monteiro, M A, Rodrigues, Ligia M C S, Curado, E M F
• Format: Article
• Language: English
• Subjects: Band structure of solids; Construction; Density; exact solutions; Functions (mathematics); Hamiltonian functions; Mathematical analysis; non-self-adjoint Hamiltonians; position dependent mass; Product data management; Schroedinger equation
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/49/12/125203

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.