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Springer eBooks

The Matrix Perturbation Method in Quantum Mechanics

Francisco Soto, Braulio Misael Villegas-Martínez, H. M. Moya-Cessa

Published Jan 1, 2023• 2 citations
Publisher
DOI
10.1007/978-3-031-48546-6

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In this paper we propose a new method of nonlinear transformation(Eigen-Mapping) to treat the eigenvalue problems in quantum mechanics as examples.The basic idea is to make the matrix of some operator we concerned diagonalization through introduction of small undetermined parameters and establishment equations for matrix elements.These problems can be devided into two types based on that there are or not 1st order approximate solution for them(in sense of the perturbation theory).In either type we consider also two cases of degenerate and nondegenerate.To obtain solution or approximate solution for eigenvalue problems,it is given the procedures of the method and usable formulas exhaustively.Seven examples in this paper are a common sight in general course of quantum mechanics.The results from them are compared with that from other approximate methods(for examples: perturbation method and variation method).After checking the results obtained,we can see that this new method has many advantages.
The original version of matrix mechanics is little used in applications nowadays. In that early theory position, momentum and other quantities were represented by infinite matrices, whereas many modern applications of quantum mechanics represent such observables as operators. Infinite matrices are unwieldy to use and have some disturbing properties, e.g. they do not obey the associative rule. By far the most commonly appearing matrix in quantum mechanics is the matrix of the energy operator as set up in some finite basis of trial functions. The procedure has been used for perturbed oscillator problems; the recurrence relation approach is particularly simple when the matrix has only one non-zero element beyond the diagonal element in each row. It seems to the author that there are still many hybrid methods which need investigation.