Analytic Results in the Position-Dependent Mass Schrödinger Problem

2013/06/30 by M. S. Cunha, H. R. Christiansen
Mathematics · Physics and Astronomy · #Class (philosophy) #Effective mass (spring–mass system) #Eigenfunction #Eigenvalues and eigenvectors #Geometry #Hyperbolic function #Hypergeometric function #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Physics #Position (finance) #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Schrödinger equation #Space (punctuation) #Tangent #hep-th #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1088/0253-6102/60/6/02

published as Commun. Theor. Phys. vol. 60, No. 6, pp. 642-650, 2013 · Some typos corrected. Version to appear in Comm. Theor. Phys. (2013)

arxiv created 2013/11/30 · openalex publication_date 2013/12/15 · arxiv updated 2014/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We investigate the Schrödinger equation for a particle with a nonuniform solitonic mass density. First, we discuss in extent the (nontrivial) position-dependent mass V(x) = 0 case whose solutions are hypergeometric functions in tanh 2 x. Then, we consider an external hyperbolic-tangent potential. We show that the effective quantum mechanical problem is given by a Heun class equation and find analytically an eigenbasis for the space of solutions. We also compute the eigenstates for a potential of the form V(x) = V 0 sinh 2 x.

Citations

• Scattering states of a particle, with position-dependent mass, in a \mathcal P\mathcal T-symmetric heterojunction
• Oscillator-Morse–Coulomb mappings and algebras for constant or position-dependent mass
• Classes of exact solutions to the Teukolsky master equation
• On reducing the Heun equation to the hypergeometric equation
• A singular position-dependent mass particle in an infinite potential well