Solutions to position-dependent mass quantum mechanics for a new class of hyperbolic potentials

Abstract

We analytically solve the position-dependent mass (PDM ) 1D Schrödinger equation for a new class of hyperbolic potentials V p q (x) = −V0 sinh p x cosh q x , p = −2, 0, . . . q [see C. A. Downing, J. Math. Phys. 54 072101 (2013)] among which several hyperbolic single-and double-wells. For a solitonic mass distribution, m(x) = m0 sech 2 (x), we obtain exact analytic solutions to the resulting differential equations. For several members of the class, the quantum mechanical problems map into confluent Heun differential equations. The PDM Poschl-Teller potential is considered and exactly solved as a particular case.

Energy eigenfunctions for position-dependent mass particles in a new class of molecular Hamiltonians

“…( 1) have three possible phases: hyperbolic singlewells, hyperbolic double-wells and hyperbolic triple-wells. Since we have already analyzed in detail the first two situations, among which the PDM Poschl-Teller [21] and PDM Manning potentials respectively, we now focus on the triple-well case which oblige the three terms.…”

“…For some relevant mass distributions some phenomenological potentials have been solved in recent years. [21][22][23][24][25][26][27][28] The origin of the PDM approximation can be traced back in the domain of solid state physics. [29][30][31][32][33][34][35][36] For instance, the dynamics of electrons in semiconductor heterostructures has been tackled with an effective mass model related to the envelope-function approximation.…”

“…The number of solvable potentials in ordinary quantum mechanics is in fact limited, but when we assume the particle mass has a nontrivial space distribution the mathematical difficulties grow even more. For some relevant mass distributions some phenomenological potentials have been solved in recent years [21][22][23][24][25][26][27][28].…”

Approximate Solutions of Dirac Equation with Hyperbolic-Type Potential

“…The confluent Heun functions HeunC(α, β, γ, δ, η, y) are convergent within |y| < 1, and asymptotic behaviour at y = 1 is not known [21,[23][24][25][26][27][28]. So, by expanding the solution about the other singular point we could obtain an analytical continuation for the Heun function.…”

“…We present here the approximate analytical solutions of Dirac equation for the case where S(r) ≃ −V (r) corresponding to pseudospin symmetry for a potential field which could be written in a exponential form. This hyperbolic-type potential has been studied for the non-relativistic case [21,22], and the effects of position-dependent mass on spectrum have also been computed [23]. The explicit form of the potential (Fig.…”