Dirac Notation and Transformation Theory

Norman J. M. Horing

Published Jul 27, 2017• 0 citations

Abstract

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.

DOI: 10.1093/oso/9780198791942.003.0001

Related papers

Basic formalism

Bipin R. Desai

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.

Cambridge University Press eBooksDec 3, 20090 citations

Some Elements of the Transformation Theory

Ahmad Amer

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

CRC Press eBooksNov 5, 20250 citations

The final synthesis of quantum mechanics: the “transformation theory” and Dirac notation

Robert Golub, S. K. Lamoreaux

Abstract The mathematics underpinning quantum mechanics is reviewed and discussed at length. The Sturm-Liouville theory of linear differential equations is presented. It is shown that the solutions form a complete set of orthogonal functions with real eigenvalues, and the magnitude of the eigenvalues form an ascending series beginning from a minimum. The formulation of the Dirac bra-ket notation and its meaning and applications is developed. We examine the idea of Hilbert space and the general abstract form of the theory as developed by Dirac and Jordan, introducing the Dirac notation which will be used in the rest of the book.

Feb 28, 20230 citations

Wavefunctions

Gary E. Bowman

This chapter focuses on a basic but systematic treatment of wavefunctions. By comparison with a spin 1/2 system, the concept of a wavefunction as a collection of coefficients is developed, followed by discussions of eigenstates, operators, and representations in the context of wavefunctions. Transformation from position to momentum representation is effected by a Fourier transform. This is developed next, and illustrated for a Gaussian. The discussion of phases and time evolution in earlier chapters is now applied to wavefunctions, and the time evolution of a free particle discussed. The concept of a wavepacket and its suitability for representing particles follows. A topic of some subtlety — and one which is discussed in detail — is the use of bra-ket (Dirac) notation for wavefunctions. The chapter closes with a short epilogue.

Oxford University Press eBooksNov 1, 20070 citations

Dirac Bra-Ket Notation

Z. Ficek

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 Â 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.

Mar 3, 20170 citations

Chapter 4

Michael A. Parker

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.

CRC Press eBooksOct 3, 20180 citations