An exactly solvable Schrödinger equation with finite positive position-dependent effective mass

Abstract

The solution of the one-dimensional Schrödinger equation is discussed in the case of position-dependent mass. The general formalism is specified for potentials that are solvable in terms of generalized Laguerre polynomials and mass functions that are positive and bounded on the whole real x axis. The resulting four-parameter potential is shown to belong to the class of “implicit” potentials. Closed expressions are obtained for the bound-state energies and the corresponding wave functions, including their normalization constants. The constant mass case is obtained by a specific choice of the parameters. It is shown that this potential contains both the harmonic oscillator and the Morse potentials as two distinct limiting cases and that the original potential carries several characteristics of these two potentials. Possible generalizations of the method are outlined.

Semiconductor Quantum Wells with BenDaniel-Duke Boundary Conditions and Janus Nanorods

“…If the mass is position dependent, according to (2), the eigenvalue equations obtained from the Schrodinger equations, using BenDaniel-Duke boundary conditions have the form:…”

“…1 in [1]). Recent papers give explicit methods to obtain explicit solutions of the Schrödinger equation with PDM, for various forms of this dependence and for several classes of potentials [2][3][4].…”

Generalized Laguerre polynomials with position-dependent effective mass visualized via Wigner’s distribution functions

“…Schrödinger equation (SE) endowed with position-dependent effective mass (PDEM) has received a growing interest on behalf of physicists during the last decade [1,2,3,4,5,6,7,8,9,10,11,12]. Its solutions has been found to be very useful in describing, physically, the properties of the quantum dynamics of electrons in condensed matter physics as well as related fields of physics [13,14,15,16].…”

$\mathcal {CPT}$ CPT -conserved effective mass Hamiltonians through first and higher order charge operator $\mathcal {C}$C in a supersymmetric framework

“…In the framework of supersymmetric (SUSY) quantum mechanics [35][36][37][38], an underlying anticommutator K of the supercharges Q and Q can be explicitly constructed by specifying the following representation…”