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American Journal of Physics

Operator techniques in three dimensions

Charles Ray Smith

Published Oct 1, 1976• 3 citations
Publisher

Abstract

Dirac bra and ket notation is developed for three-dimensional space. A number of transformations, closure relations, and projection operators are discussed in some detail. The formal results are applied to examples pertaining to homogeneous isotropic and anisotropic systems, as well as to sums over photon polarization degrees of freedom. Most of the discussion is tailored for the student before his exposure to operator methods of quantum mechanics.

DOI
10.1119/1.10247

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Dirac bra and ket notation is developed for three-dimensional space. A number of transformations, closure relations, and projection operators are discussed in some detail. The formal results are applied to examples pertaining to homogeneous isotropic and anisotropic systems, as well as to sums over photon polarization degrees of freedom. Most of the discussion is tailored for the student before his exposure to operator methods of quantum mechanics.
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.
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.
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.
According to Dirac's bra-ket notation, in an inner-product space, the inner product ⟨x ∣ y⟩\langle x\,|\,y\rangle⟨x∣y⟩ of vectors x,yx,yx,y can be viewed as an application of the bra ⟨x∣\langle x|⟨x∣ to the ket ∣y⟩|y\rangle∣y⟩. Here ⟨x∣\langle x|⟨x∣ is the linear functional ∣y⟩↦⟨x ∣ y⟩|y\rangle \mapsto \langle x\,|\,y\rangle∣y⟩↦⟨x∣y⟩ and ∣y⟩|y\rangle∣y⟩ is the vector yyy. But often -- though not always -- there are advantages in seeing ∣y⟩|y\rangle∣y⟩ as the function a↦a⋅ya \mapsto a\cdot ya↦a⋅y where aaa ranges over the scalars. For example, the outer product ∣y⟩⟨x∣|y\rangle\langle x|∣y⟩⟨x∣ becomes simply the composition ∣y⟩∘⟨x∣|y\rangle \circ \langle x|∣y⟩∘⟨x∣. It would be most convenient to view kets sometimes as vectors and sometimes as functions, depending on the context. This turns out to be possible. While the bra-ket notation arose in quantum mechanics, this note presupposes no familiarity with quantum mechanics.
We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.