Dirac introduced a very useful notation of state vectors Ψ i in terms of "bra" vectors. This notation allows to make the formal expressions of quantum physics more transparent and easier to manipulate. The bra-ket notation extends to action of operators on state vectors. The projection operator and the completeness relation are very important quantities in quantum physics. They are very useful in calculations and are often employed to represent an arbitrary operator and a wave function (state) in terms of orthonormal states. To illustrate the procedure of representations, this chapter considers an arbitrary operator  and shows how to represent the operator in terms of projection operators of known orthonormal states. The example in the chapter illustrates the concept of orthonormal states. It presents the application of the completeness relation. Using the completeness relation, that matrix elements of the product operator in the same orthonormal basis can be found from the multiplication of the matrix elements.
Dirac introduced a very useful notation of state vectors Ψ i in terms of "bra" vectors. This notation allows to make the formal expressions of quantum physics more transparent and easier to manipulate. The bra-ket notation extends to action of operators on state vectors. The projection operator and the completeness relation are very important quantities in quantum physics. They are very useful in calculations and are often employed to represent an arbitrary operator and a wave function (state) in terms of orthonormal states. To illustrate the procedure of representations, this chapter considers an arbitrary operator  and shows how to represent the operator in terms of projection operators of known orthonormal states. The example in the chapter illustrates the concept of orthonormal states. It presents the application of the completeness relation. Using the completeness relation, that matrix elements of the product operator in the same orthonormal basis can be found from the multiplication of the matrix elements.
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
Abstract Chapter 1 opens with a brief review of some basic features of quantum mechanics, including the Schrödinger equation, linear and angular momentum and the theory of the hydrogenic atom: It also includes complete orthonormal sets of eigenfunctions, the translation operator, current, spin, equation of continuity, gauge transformation, determinant & permanent multiparticle energy eigenfunctions for noninteracting particles and the Pauli exclusion principle. Attention is then focused on Dirac bra-ket notation and complete sets of commuting observables. In this connection, representations and transformation among representations are discussed in detail for the Schrödinger system state vector and the eigenstates, as well as bra-ket matrix elements of operators. Finally, Schwinger’s interpretation of ket-bra matrix operator structures (Schwinger “Measurement Symbols”) in terms of annihilation and creation of systems in eigenstates is introduced.
We summarize below some of the postulates and definitions basic to our formalism, and present some important results based on these postulates. The formalism is purely mathematical in nature with very little physics input, but it provides the structure within which the physical concepts that will be discussed in the later chapters will be framed. State vectors It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex, linear vector space. This vector may represent a free particle or a particle bound in an atom or a particle interacting with other particles or with external fields. It is much like a vector in ordinary three-dimensional space, following many of the same rules, except that it describes a very complicated physical system. We will be elaborating further on this in the following. The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac. A physical state in this notation is described by a “ket” vector, |〉, designated variously as |α〉 or |ψ〉 or a ket with other appropriate symbols depending on the specific problem at hand. The kets can be complex. Their complex conjugates, |〉 * , are designated by 〈| which are called “bra” vectors. Thus, corresponding to every ket vector there is a bra vector. These vectors are abstract quantities whose physical interpretation is derived through their so-called “representatives” in the coordinate or momentum space or in a space appropriate to the problem under consideration.
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 .
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.
In this article, we discard the bra-ket notation and its correlative definitions, given by Paul Dirac. The quantum states are only described by the wave functions. The fundamental concepts and definitions in quantum mechanics is simplified. The operator, wave functions and square matrix are represented in the same expression which directly corresponds to the system of equations without additional introduction of the matrix representation of operator. It can make us to convert the operator relations into the matrix relations. According to the relations between the matrices, the matrix elements will be determined. Furthermore, the first order differential equations will be given to find the solution of equations. As a result, we unified the descriptions of the matrix mechanics and the wave mechanics.