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arXiv:2008.03187 · European Journal of Physics

Do bras and kets have dimensions?

Claude Semay, Cintia T. Willemyns

Published Sep 22, 2020• 1 citation
arXivHTMLPDF

Abstract

Abstract The bra and ket notation introduced by Dirac and the dimensional analysis are two powerful tools for the physicist. Curiously, almost nothing is said about the connections between these two topics in the literature. We show here that the notion of dimensions is relevant for bras and kets. This could help students to grasp a better comprehension of this abstract notation.

DOI
10.1088/1361-6404/abbab1

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Abstract The bra and ket notation introduced by Dirac and the dimensional analysis are two powerful tools for the physicist. Curiously, almost nothing is said about the connections between these two topics in the literature. We show here that the notion of dimensions is relevant for bras and kets. This could help students to grasp a better comprehension of this abstract notation.
When teaching quantum mechanics, little consideration is given to the dimensional aspect of wave functions and the Dirac notation. The introduction of the Dirac notation, with its bra and ket vectors, is a powerful tool for describing quantum states. However, the question of whether the Dirac notation has dimensions is rarely discussed. It seems that it is commonly assumed that the Dirac notation is merely a dimensionless mathematical symbol. In this article, we will discuss the dimensional aspect of wave functions and the Dirac notation, which will help students develop a deeper understanding of the meaning behind quantum mechanical symbols.
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.
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.
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.