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 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.
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).
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.
The quantum theoretical description of microscopic systems is based on differential operators and the solution of the underlying eigenvalue equations. This approach is called Schrödinger approach or wave mechanics. Heisenberg developed another equivalent approach called matrix mechanics. In matrix mechanics all observables are represented by matrices. All possible eigenvalues of a matrix representing an observable are the possible outcomes of experimental measurement on that observable. This chapter gives the preliminary concepts relative to the use of the matrix method. It presents the various elements of matrix mechanics and discusses the matrix representation of operators of certain systems. The chapter introduces Dirac's notation of state vectors, Hilbert space, projection operator, displacement operators and the generators of translations. The theory of quaternions has found applications in classical mechanics, quantum mechanics and relativity. It has been used in aerospace applications, flight simulators, computer graphics, animation, visualization, fractals and virtual reality.
Linear algebra provides the mathematical foundation of quantum mechanics by rep-resenting physical states as vectors in complex Hilbert spaces and observables as linear operators. The concepts of vector spaces, basis, orthogonality, inner products, and matrix transformations enable a rigorous description of quantum systems. This paper presents a concise overview of the linear algebraic principles underlying quantum state representa-tion. The study discusses Hilbert spaces, Dirac notation, state normalization, superposition, the Born probability rule, Hermitian operators, and quantum measurement. Furthermore, the role of tensor products and matrix representations in quantum computing is briefly highlighted. An illustrative example demonstrates the application of these mathematical concepts in representing and analysing a quantum state. The paper emphasizes that linear algebra not only provides the language of quantum mechanics but also serves as the compu-tational framework for modern quantum technologies such as quantum computing, quantum information, and quantum communication. Keywords: Linear Algebra, Quantum Mechanics, Hilbert Space, Quantum State, Hermitian Operator, Superposition, Quantum Computing.
The content of this chapter may serve as yet another supplemental topic to meet the needs and interests beyond those of a usual course curriculum. Here we shall present an over-simplified, but hopefully totally transparent, description of some of the fundamental ideas and concepts of quantum mechanics, using a pure linear algebra formalism. Vectors in ℂ n and Dirac bracket Consider the vector space ℂ n , consisting of column vectors, and use { e 1 , …, e n } to denote the standard basis of ℂ n . For u , υ ∈ ℂ n with recall that the Hermitian scalar product is given by so that { e 1 , …, e n } is a unitary basis, satisfying ( e i , e j ) = δ ij , i, j = 1, …, n . In quantum mechanics, it is customary to rewrite the scalar product (9.1.2) in a bracket form, 〈 u | v 〉. Then it was Dirac who suggested to view 〈 u |υ〉 as the scalar pairing of a ‘bra’ vector 〈 u | and a ‘ket’ vector |υ〉, representing the row vector u † and the column vector υ. Thus we may use | e 1 〉, …, | e n 〉 to denote the standard basis vectors of ℂ n and represent the vector u in ℂ n as Therefore the bra-counterpart of | u 〉 is simply given as As a consequence, the orthonormal condition regarding the basis { e 1 , …, e n } becomes and the Hermitian scalar product of the vectors | u 〉 and |υ〉 assumes the form For the vector | u 〉 given in (9.1.3), we find that a i = 〈 e i | u 〉, i = 1, …, n . Now rewriting | u 〉 as and inserting (9.1.7) into (9.1.8), we obtain which suggests that the strange-looking ‘quantity’, should naturally be identified as the identity mapping or matrix, which readily follows from the associativity property of matrix multiplication. Similarly, we have Thus (9.1.10) can be applied to both bra and ket vectors symmetrically and what it expresses is simply the fact that | e 1 〉, …,| e n form an orthonormal basis of ℂ n .
Chapter 3 introduces the student to the matrix formulation of quantum mechanics, pioneered by Heisenberg in collaboration with Born and Jordan. It provides the basic formal elements of the Heisenberg description in terms of operators and their matrix representation, derives the Heisenberg inequality, and applies the formalism to the harmonic oscillator as a central and particularly simple quantum problem. It introduces the state vectors and their Hilbert space, and concludes with a brief discussion hinting at the physical significance of the replacement of classical continuous dynamical variables with operators as a manifestation of the coupling of the particle to the radiation field.