The bases used in quantum mechanics, in general, are orthonormal. The relations in orthonormal basis vectors are simpler than those in nonorthogonal basis vectors. Also, the operators that represent different physical quantities are Hermitian and the eigenvectors of these operators belonging to different eigenvalues are orthonormal to each other. But, in many cases, such as sets of screened hydrogenic wavefunctions, nonorthogonal sets of basis vectors occur. It is, therefore, necessary to consider the representation of vectors and operators in nonorthogonal bases. Making use of the set of vectors reciprocal to the given set of nonorthogonal base vectors, the expression for the norms of vectors and for the matrix elements has been found. Also, adjoint and Hermitian matrices in nonorthogonal bases have been worked out using a set of reciprocal basis vectors.
The bases used in quantum mechanics, in general, are orthonormal. The relations in orthonormal basis vectors are simpler than those in nonorthogonal basis vectors. Also, the operators that represent different physical quantities are Hermitian and the eigenvectors of these operators belonging to different eigenvalues are orthonormal to each other. But, in many cases, such as sets of screened hydrogenic wavefunctions, nonorthogonal sets of basis vectors occur. It is, therefore, necessary to consider the representation of vectors and operators in nonorthogonal bases. Making use of the set of vectors reciprocal to the given set of nonorthogonal base vectors, the expression for the norms of vectors and for the matrix elements has been found. Also, adjoint and Hermitian matrices in nonorthogonal bases have been worked out using a set of reciprocal basis vectors.
This chapter illustrates how one can obtain a matrix representation of a state vector, an operator, and an eigenvalue equation in a basis of orthonormal states. It helps to learn that matrix representations are very useful in calculations of the properties of operators, because algebra of matrices is very simple and completely developed. Readers familiar with the algebra of matrices agree that mathematical operations on matrices such as multiplication by a scalar, addition, subtraction, multiplication and diagonalization are simple and easy to perform. In fact, the application of the algebra of matrices to quantum physics results from the direct relations between operators and matrices that any mathematical relation that holds between operators also holds between the corresponding matrices. The eigenvalues and eigenvectors of the matrix representing a given operator  in the basis of orthonormal states are the same as the eigenvalues and eigenvectors of Â.
Linear algebra is the natural mathematical language of quantum mechanics. For this reason, the present chapter starts with a review of Hilbert spaces for vectors and operators. We introduce vector and Hilbert spaces along with inner products and metrics. The Dirac notation is developed for the Euclidean vector spaces as a starting point for the concepts of complete orthonormal sets of vectors, closure, dual vector spaces, and adjoint operators. The Dirac delta function in various forms and the principal part is introduced in Appendix 5 as an essential tool. The chapter turns to the main use of Dirac notation for function spaces; the concepts of norm, inner product, and closure are discussed. Fourier, Cosine, and Sine series are discussed as examples of expansions in complete orthonormal sets of functions. Although Hilbert spaces are interesting mathematical objects with important physical applications, the study of linear algebra remains incomplete without a study of linear operators (i.e., linear transformations). In fact, the set of linear transformations itself forms a vector space and therefore has a basis set. The basis set for the operator is linked with the basis sets for the spaces that it operates between. The linear operator can be discussed as an abstract operator or through an isomorphism as a matrix or as a generalized expansion in operator space. A Hermitian (a.k.a., self-adjoint) operator produces a basis set within a Hilbert space. The basis set comes from the eigenvector equation for the particular operator. The fact that a Hermitian operator produces a complete set (of orthornomal vectors) has special importance for quantum mechanics. Observables such as energy or momentum correspond to Hermitian operators. Complete sets make it possible to represent every possible result of a measurement of the observable by an object (vector) in the theory. The Hermitian operators have real eigenvalues which represent the results of the measurement.
This chapter introduces the representation of quantum states and operators in an orthonormal basis associated with a linear Hermitian operator, forming the foundation of matrix mechanics. The transformation between different representations is described, highlighting how operators and state vectors change under basis transformations. The projection operator is defined, and the closure (completeness) relation is expressed in terms of projection operators on the basis states. The Dirac bra–ket notation is introduced and thoroughly explained to unify representation theory. Detailed treatments of the coordinate and momentum representations are given, including the transformation between them via Fourier analysis. Finally, explicit forms of the coordinate and momentum operators acting on wavefunctions in the coordinate space are derived. Metadata/Keywords: matrix mechanics, representation theory, Hermitian operator, projection operator, closure relation, Dirac bra–ket notation, coordinate and momentum representations
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.
This chapter provides a simple general introduction to some basic mathematics necessary for a subsequent discussion of Lie algebras, operator methods, and their applications to theoretical physics and chemistry. It also provides some useful mathematical tools to derive important theoretical results and expressions in a simple and elegant way. In quantum mechanics the state of a system is represented by an element of a particular vector space, in which one defines an appropriate scalar or inner product, known in mathematics as Hilbert space. Hermitian operators with a complete set of eigenvectors belonging to that state space represent physical observables. Moreover, in certain cases it proves useful to work with vector spaces formed by linear operators. Hermitian operators are particular cases of linear operators and physical observables are particular cases of Hermitian operators. In many physical applications of operator methods one needs to reorder products of exponential operators or to rewrite one such operator as a product of simpler ones.
Abstract Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by Hermitian linear operators that act on state vectors that belong to a Hilbert vector space. This chapter analyzes problems that deal with the basic concepts of the theory of linear operators; eigenfunctions, eigenvalues, mean values; the projection operators; quantum-mechanical representations of operators and wave-functions; and unitary operators.