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Boundary least squares method for the solution of schrödinger's equation in quantum wires
Author(s) -
Sirenko Yu. M.,
Mitin V.
Publication year - 1996
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/pssb.2221960108
Subject(s) - mathematics , eigenfunction , piecewise , mathematical analysis , rayleigh quotient , residual , schrödinger equation , quadrilateral , boundary value problem , wave function , minification , piecewise linear function , square (algebra) , eigenvalues and eigenvectors , quantum mechanics , geometry , physics , mathematical optimization , finite element method , algorithm , thermodynamics
We present a method for the approximate semianalytical calculation of wave functions and eigenenergies in systems consisting of domains with knownbulk solutions of the corresponding Schrödinger equation (e.g. with piecewise constant potential). The trial wave function is written as a normalized linear combination of several bulk solutions for the given energy. The coefficients of the linear combination are found by minimization of the integral of square mismatch along the boundaries. Mathematically the problem is equivalent to the minimization of the Rayleigh quotient or solution of the generalized eigenproblem for the vector of coefficients. The value of the residual provides an estimation of the accuracy of the results and gives thepossibility to choose an optimal set of trial functions. We illustrate the use of the method by calculation of eigenfunctions of infinitely long triangular and quadrilateral wires.