Abstract
This chapter defines matrices and their algebra and illustrates their direct relationship to certain operators. The operators in question are those that form the basis of the subject of quantum mechanics as well as those employed in the application of group theory to the analysis of molecular vibrations and the structure of crystals. The operators that are involved in quantum mechanics are linear. The chapter explores the Hermitian operators and their eigenvalues. All operators of interest in quantum mechanics are Hermitian. Because the eigenvalues correspond to physically observable quantities, they are real and their operators are Hermitian. A matrix is an array of numbers and for most practical purposes it is rectangular. The chapter explains certain properties of matrices and defines several special matrices. The Hermitian matrix is of particular importance in quantum-mechanical applications. The chapter presents the study of determinants and its properties. A determinant is a very special case in which a given square matrix has a specific numerical value. The method of Jacobians is certainly the most widely applied and a systematic method of deriving the partial derivatives is given in the chapter. The formulas associated with obtaining the vector product by ordinary matrix multiplication, partitioning of matrices, and matrix formulation of the eigenvalue problem is provided. A simple eigenvalue problem is demonstrated by an example of two coupled oscillators. The chapter explains how the matrix method can be applied in quantum mechanics with the help of harmonic oscillator. The matrix formulation is more abstract than Schrodinger's method and its success often depends on judicious guesses.
Published Version
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