From here, we take up the task of reformulating quantum mechanics in the language of inner product space. We start by introducing vector space through the familiar example of a set of finite dimensional, complex, column matrices under addition and multiplication by scalars. The inner product is then introduced as a dot product generalized to complex vectors leading to the idea of an inner product space . This is adequate to show how the concept of a quantum state can be described by a normalized vector in an inner product space so that the state space can be identified with the inner product space. Amplitudes are associated with inner products , and observables with orthonormal bases in this space. Hence the laws of quantum mechanics are reformulated. We then point out that the now reformulated laws are independent of the space of description and the reader is urged to contemplate an universality which would reflect the independence with respect to the representation space . This creates the stage for the introduction of the abstract inner product space and the formulation of quantum mechanics in this setting. The chapter concludes with this description. The transformed laws are now called postulatesof quantum mechanics .
From here, we take up the task of reformulating quantum mechanics in the language of inner product space. We start by introducing vector space through the familiar example of a set of finite dimensional, complex, column matrices under addition and multiplication by scalars. The inner product is then introduced as a dot product generalized to complex vectors leading to the idea of an inner product space . This is adequate to show how the concept of a quantum state can be described by a normalized vector in an inner product space so that the state space can be identified with the inner product space. Amplitudes are associated with inner products , and observables with orthonormal bases in this space. Hence the laws of quantum mechanics are reformulated. We then point out that the now reformulated laws are independent of the space of description and the reader is urged to contemplate an universality which would reflect the independence with respect to the representation space . This creates the stage for the introduction of the abstract inner product space and the formulation of quantum mechanics in this setting. The chapter concludes with this description. The transformed laws are now called postulatesof quantum mechanics .
There is a notational discrepancy in how mathematicians and physicists treat Hilbert spaces. In physics, one often works with finite-dimensional Hilbert spaces treated as ℂ n , and hence wants to write the Hermitian inner product as (x, y) = x ∗ y. This definition dictates the use of a Hermitian inner product that is antilinear in the first argument, the convention followed, for example, in Reed & Simon [246]. This is also the choice adopted in Dirac’s bracket notation, whose usage in quantum mechanics is very widespread. Our notation is chosen to extend-as closely as possible-the traditional notation of standard finite-dimensional matrix algebra to arbitrary complex inner product spaces and associated linear operators. In matrix algebra, column vectors and the corresponding matrices with one column are identical objects, row vectors are the linear functionals, and the adjoint is the conjugate transpose. For example, ℍ = ℍ × = ℂ n is the space of column vectors of size n, the dual space ℍ ∗ is the space of row vectors of size n, and the operator product ϕ ∗ ψ of a row vector ϕ ∗ and a column vector ψ is the standard Hermitian inner product of the column vectors ϕ and ψ. We use Greek lower case letters to write vectors, thus emphasizing their intended use as quantum state vectors in quantum mechanics. On the other hand, mathematicians working on reproducing kernel Hilbert spaces use an inner product (x, y) antilinear in the second argument, related to the physicist’s inner product (x, y) by (x, y) =( ). This is the convention followed, for example, in Rudin [252]. Although the two ways of defining the inner product lead to fully equivalent theories, all details look a bit different, a fact that has to be taken into account when reading the literature on the subject. For example, in the description based on the physical tradition, it is preferable to work with the antidual space in place of the dual space used in the mathematical tradition. In functional analysis, linear operators in Hilbert spaces are usually considered each with their own domain. But many computations in quantum mechanics require the consideration of algebras of operators with a common domain. The latter is a Euclidean space, a dense subspace of a Hilbert space. This space and its antidual play in many respects a more basic role in quantum physics than the Hilbert space itself. Therefore, and to avoid possible confusion caused by the different traditions, we give in the present chapter a self-contained introduction to Euclidean spaces and their associated spaces. All proofs are carried out in detail.
Vector spaces are the mathematical structures in which the state vectors of quantum computation live. When endowed with an inner product, a vector space becomes an inner product space. Of particular relevance are complex inner product spaces. An objective of this chapter is to provide a formal definition of a vector space and give several examples. You will see that, in this context, a ‘vector’ is a much more general object than vectors such as those in https://www.w3.org/1998/Math/MathML" display="inline"> ℝ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math6_1.tif "/> and https://www.w3.org/1998/Math/MathML" display="inline"> ℝ 3 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math6_2.tif "/> . However, the already familiar operations of vector addition and scalar multiplication are essential ingredients of all vector spaces. We demonstrate how vectors in a vector space can be constructed using a set of vectors called a basis and see how vectors can be written in terms of different bases. Of particular importance are orthonormal bases in which the basis vectors are mutually orthogonal and normalised. Finally, we introduce cosets and explain how a quotient space is constructed.
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 .
With the postulates already formulated in the language of an abstract inner product spaces, this chapter shows how an observable can be naturally expressed as a linear Hermitian operator acting on the inner product space where the states live. We start by introducing linear operators acting on vector spaces and their representations. Next, we discuss the systems of eigenvalues and eigenvectors associated with such operators. The important case of Hermitian operator and the properties of their eigenvalues and eigenvectors, relevant to quantum mechanics, are introduced. After developing the necessary background in linear algebra, the connection of Hermitian operators with quantum observables is established. We then proceed to explain what this connection actually accomplishes. In particular, we show how it translates the target questions of quantum mechanics into an eigenvalue problem. This makes it evident what the nature of the input to a quantum theory should be. This input enters the formalism through what are known as “quantization rules”. I discuss this in the last section of this chapter. An illustration using a disguised spin-half system winds up the chapter.
We review the postulates of quantum mechanics with respect to the representation of physical states and measurable quantities, their time evolution, and the interpretation of measurements. We first formulate the postulates in terms of wave functions and differential operators, and then reformulate them in the abstract Hilbert space of state vectors, using Dirac’s notations. Improper states subject to Dirac’s delta normalization are introduced, and the space of physical states is extended to include them. The postulates are rationalized by associating each Hermitian linear operator with a complete orthonormal system of its eigenvectors, where measurement probabilities depend on the projections of these eigenvectors on the system’s state vector. Particularly, wave functions are identified as projections of state vectors on the position operator eigenstates. State vectors representing multidimensional systems are formulated as tensor products of vectors in their subspaces. Finally, we address the general uncertainty relations in simultaneous measurements of different observables.
Abstract Quantum mechanics is mathematically formulated in terms of a Hilbert space, which is a vector space with a defined inner product. Linear algebra is the study of vector spaces and the (linear) operations thereon, and so it follows that having a solid grasp of the basics of linear algebra is essential for studying quantum mechanics and, therefore, quantum computation. All the linear algebra prerequisites necessary to understand the theoretical results presented in the remainder of the book are detailed in this chapter. The chapter also acts as a nomenclature reference for the book, by providing all the notations that are used thereafter.