Supersymmetric Approach to Exactly Solvable Systems With Position-Dependent Effective Masses

Abstract

We discuss the relationship between exact solvability of the Schrödinger equation with a position-dependent mass and the ordering ambiguity in the Hamiltonian operator within the frame of supersymmetric quantum mechanics. The one-dimensional Schrödinger equation, derived from the general form of the effective mass Hamiltonian, is solved exactly for a system with exponentially changing mass in the presence of a potential with similar behaviour, and the corresponding supersymmetric partner Hamiltonians are related to the effective-mass Hamiltonians proposed in the literature.

-fold supersymmetry in quantum systems with position-dependent mass

“…For references, see e.g., Refs. [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and those cited therein. Due to the fact that the position-dependent mass m(q) does not commute with the momentum operator p = −id/dq, ambiguity arises in defining a quantum kinetic operator which is formally Hermitian and reduces to the classical kinetic term T = p 2 /2m(q).…”

“…A different choice of the parameters results in a different correction to the original potential profile V (q), and the above Hamiltonian always has the following form: Hence, the typical investigations into (quasi-)exact solvability of PDM quantum systems consist in finding simultaneously a pair of an effective potential U(q) and a mass function m(q) for which the PDM Hamiltonian (1.2) admits (a number of) exact eigenfunctions in closed form. Up to now, two different methods have been frequently employed, namely, coordinate transformations including point canonical transformations [1,2,5,6,7,8,9,12,16,18,20,25], and supersymmetric methods [3,4,7,8,11,13,14,19,20,21,22,24,25]. The latter approaches were also applied to many-body PDM quantum systems [26].…”

Explicit solutions for N-dimensional Schrödinger equations with position-dependent mass

“…Recently, some researches have been devoted to the analysis of the classification of quantum systems with position-dependent mass regarding their exact solvability [3][4][5], and the references therein]. On a similar basis, Plastino and his co-workers [8] applied an approach within the supersymmetric quantum mechanical framework, for the case α = γ = 0, to such systems and succeed to show that some one-dimensional systems with non-constant mass have a supersymmetric partner with the same effective mass.…”

A squeeze-like operator approach to position-dependent mass in quantum mechanics

“…Among them, one may mention the Morse and Coulomb potentials [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Moreover, it has been recently shown [19] that to lowest order of perturbation theory, there exists a whole class of Hermitian position-dependent-mass Hamiltonians that are associated with pseudo-Hermitian Hamiltonians.…”