Exact Solutions of Effective-Mass Schrödinger Equations

Abstract

We outline a general method for obtaining exact solutions of Schrödinger equations with a position dependent effective mass and compare the results with those obtained within the frame of supersymmetric quantum theory. We observe that the distinct effective mass Hamiltonians proposed in the literature in fact describe exactly equivalent systems having identical spectra and wave functions as far as exact solvability is concerned. This observation clarifies the Hamiltonian dependence of the band-offset ratio for quantum wells.

-fold supersymmetry in quantum systems with position-dependent mass

“…(4.13), (4.24), and (4.27), we obtain the operator P N for the type A N -fold supercharge 8) where the products of operators are ordered according to the following definition:…”

“…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].…”

Generation of Exactly Solvable Potentials of Position-Dependent Mass Schrödinger Equation from Hulthen Potential

“…Such studies use different methods known for solving constant mass Schrödinger equations or an extension of them. Point Canonical Transformation (PCT) [6][7][8][9][10], Nikiforov-Uvarov (NU) method [11][12][13], Supersymmetry (SUSY) quantum mechanics approach [14,15], Quadratic Algebra [16], Darboux Transformation (DT) [17,18] etc. are different approaches used in the study of PDM Schrödinger equations.…”

PT-Symmetric Solutions of Schrödinger Equation with Position-Dependent Mass via Point Canonical Transformation

“…Important applications are obtained in the fields of material science and condensed matter physics such as semiconductors [1], quantum well and quantum dots [2], 3 H clusters [3], quantum liquids [4], graded alloys and semiconductor heterostructures [5,6]. Recently, number of exact solutions on these topics increased [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Various solution methods are used in the calculations.…”