Chapter 3. Matrix Mechanics

Caio Lima Firme

Matrices, vectors and determinants are presented in this chapter, mainly their applications to the quantum mechanics and quantum chemistry. A very important property of the matrices is that they do not commute and this leads to the commutator relations discussed in the end of this chapter and to most important features of the quantum matrix mechanics, for example, the uncertainty principle. In this chapter, the most important properties of the matrices (trace of matrix, matrix multiplication, transpose of a matrix, symmetric and orthogonal matrix) are shown along with the methods to obtain the determinant of the matrix (Gaussian elimination and Laplace expansion). The Gauss-Jordan method to obtain the inverse matrix is presented and antissymetric matrices and its importance to quantum mechanics. Matrix diagonalization, Jacobi method of diagonlization, eigenvectors in vector space, Hermitian matrix, unitary transformation and linear transformations, characteristic equations and system of secular homogeneous linear equations are shown in sequence which are vital for several operations in the quantum wave mechanics. Several Fortran codes of linear algebra are also displayed in this chapter (174 words).

James M. Feagin

Caio Lima Firme

In this chapter shows some applications of matrix and wave mechnanics. The equivalence between matrix mechanics and wave mechanics is explored in this chapter. Another approach for this equivalence is indicated as well. The Heisenberg equation of motion is applied to the quantum virial theorem and quantum theory of atoms in molecules (QTAIM) in next section. It is introduced afterwards that the Heisenberg’s matrix mechanics is applied to the helium atom leads to the concept of quantum resonance and chemical bond. Next, the use of the matrix mechanics and wave mechanics to obtain the uncertainty principle is discussed and its derivation is also given. Subsequently, Dirac’s quantum mechanics – an amalgamation of matrix mechanics and wave mechanics – is introduced. Finally, the importance of the commutation of the quantum operators to the quantum mechanics is discussed followed by some basic quantum principles from the Copenhagen interpretation. A couple of examples and exercises is also available in this chapter.(156 words).

Arjun Sharma

This chapter deals with the matrix and vector formulation of quantum mechanics. Topics covered under the chapter are: matrices and their basic algebra, bra and ket notations, vector space and the basis vectors, inner and outer products of vectors, the orthogonalization procedure for vectors, the Schwarz inequality, linear transformation of a vector, eigenvalues and eigenvectors of a matrix, numerical methods to find eigenvalue and eigenvector, different types of matrices used in quantum mechanics, diagonalization of a matrix, the Cayley-Hamilton theorem, change of basis in a vector space and the unitary transformations, infinite-dimensional and Hilbert spaces, representation of quantum states and operators in Hilbert space, the Schrödinger equation in matrix form, assumptions of quantum mechanics in matrix formulations, the general uncertainty principle, the one dimensional harmonic oscillator, solved examples and unsolved exercises. The objectives of this chapter are to introduce the important aspects of vector and matrix algebra that have been vital in the development and the description of modern quantum mechanics and in exhibiting that the Heisenberg and Schrödinger approaches to quantum mechanics are equivalent.