Physics
Hilbert Space
Hilbert space is a mathematical concept used in quantum mechanics to describe the state of a quantum system. It is a complex vector space equipped with an inner product that allows for the representation of quantum states and operators. In physics, Hilbert space provides a framework for understanding the behavior and properties of quantum systems.
Written by Perlego with AI-assistance
Related key terms
1 of 5
11 Key excerpts on "Hilbert Space"
eBook - PDF- Werner O. Amrein(Author)
- 2009(Publication Date)
- EPFL PRESS(Publisher)
CHAPTER 1 Hilbert Spaces Hilbert Space sets the stage for standard quantum theory: the pure states of a physical system are identified with the unit rays of a Hilbert Space H and observables with self- adjoint operators acting in H. In this initial chapter we present the essential concepts and prove the basic results concerning separable Hilbert Spaces (Sections 1.1 - 1.3). In Section 1.4 we then introduce L 2 spaces, which are of special importance for quantum mechanics. This requires some familiarity with measure theory, and we include a short description of the necessary concepts from this theory. 1.1 Definition and elementary properties 1.1.1. Throughout this text a Hilbert Space means a complex linear vector space, equipped with a Hermitian scalar product, which is complete and admits a countable basis. More precisely a (separable) Hilbert Space H is defined by the four postulates (H1) - (H4) stated below: (H1) H is a linear vector space over the field C of complex numbers: With each couple {f,g} of elements of H there is associated another element of H, denoted f + g, and with each couple {α,f }, α ∈ C, f ∈ H, there is associated an ele- ment αf of H, and these associations have the following properties (where f,g,h ∈ H and α,β ∈ C): f + g = g + f f + (g + h) = (f + g) + h (1.1) α(f + g) = αf + αg (α + β)f = αf + βf (1.2) α(βf ) = (αβ)f 1 f = f. (1.3) Furthermore there exists a unique element 0 ∈ H (called the zero vector) such that 1 0 + f = f , 0 f = 0 ∀f ∈ H. (1.4) 1 Here 0 denotes the complex number α = 0. 2 Hilbert SpaceS (H2) H is equipped with a strictly positive scalar product 2 : With each couple {f,g} of elements of H there is associated a complex number (f,g), and this association has the following properties 3 : (g,f ) = (f,g) ∀f,g ∈ H (1.5) (f,g + αh) = (f,g) + α(f,h) ∀ α ∈ C, ∀f,g,h ∈ H (1.6) (f,f ) > 0 except for f = 0. (1.7) One then defines bardblf bardbl := [(f,f )] 1/2 . (1.8) (H3) H is complete: Each Cauchy sequence in H has a limit in H.
eBook - PDFThe Mathematical Language of Quantum Theory
From Uncertainty to Entanglement
- Teiko Heinosaari, Mário Ziman(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
1 Hilbert Space refresher Quantum theory, in its conventional formulation, is built on the theory of Hilbert Spaces and operators. In this chapter we go through this basic material, which is central for the rest of the book. Our treatment is mainly intended as a refresher and a summary of useful results. It is assumed that the reader is already familiar with some of these concepts and elementary results, at least in the case of finite-dimensional inner product spaces. We present proofs for propositions and theorems only if the proof itself is con- sidered to be instructive and illustrative. This gives us the freedom to present the material in a topical order rather than in the strict order of mathematical impli- cation. Good references for this chapter are the functional analysis textbooks by Conway [45], Pedersen [113] and Reed and Simon [121]. These books also contain the proofs that we skip here. 1.1 Hilbert Spaces As an introduction, before a formal definition is given, one may think of a Hilbert Space as the closest possible generalization of the inner product spaces C d to infi- nite dimensions. Actually, there are no finite-dimensional Hilbert Spaces (up to isomorphisms) other than C d spaces. The crucial requirement of completeness becomes relevant only in infinite-dimensional spaces. This defining property of Hilbert Spaces guarantees that they are well-behaved mathematical objects, and many calculations can be done almost as easily as in C d . 1.1.1 Finite- and infinite-dimensional Hilbert Spaces Let H be a complex vector space. We recall that a complex-valued function ·|· on H × H is an inner product if it satisfies the following three conditions for all ϕ, ψ, φ ∈ H and c ∈ C: 1 2 Hilbert Space refresher • ϕ|cψ + φ = c ϕ|ψ + ϕ|φ (linearity in the second argument), • ϕ|ψ = ψ |ϕ (conjugate symmetry), • ψ |ψ > 0 if ψ = 0 (positive definiteness). A complex vector space H with an inner product defined on it is an inner prod- uct space.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 5 Hilbert Space Hilbert Spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert Space , named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert Space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. In addition, ________________________ WORLD TECHNOLOGIES ________________________ Hilbert Spaces are required to be complete , a property that stipulates the existence of enough limits in the space to allow the techniques of calculus to be used. Hilbert Spaces arise naturally and frequently in mathematics, physics, and engineering, typically as infinite-dimensional function spaces. The earliest Hilbert Spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer) and ergodic theory which forms the mathematical underpinning of the study of thermodynamics. John von Neumann coined the term Hilbert Space for the abstract concept underlying many of these diverse applications. The success of Hilbert Space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert Spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert Space theory.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- White Word Publications(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 1 Hilbert Space Hilbert Spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert Space , named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert Space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. ________________________ WORLD TECHNOLOGIES ________________________ Furthermore, Hilbert Spaces are required to be complete, a property that stipulates the existence of enough limits in the space to allow the techniques of calculus to be used. Hilbert Spaces arise naturally and frequently in mathematics, physics, and engineering, typically as infinite-dimensional function spaces. The earliest Hilbert Spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer) and ergodic theory which forms the mathematical underpinning of the study of thermodynamics. John von Neumann coined the term Hilbert Space for the abstract concept underlying many of these diverse applications. The success of Hilbert Space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert Spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert Space theory.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 9 Hilbert Space Hilbert Spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert Space , named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert Space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert Spaces are required to be complete, a property that stipulates the existence of enough limits in the space to allow the techniques of calculus to be used. ________________________ WORLD TECHNOLOGIES ________________________ Hilbert Spaces arise naturally and frequently in mathematics, physics, and engineering, typically as infinite-dimensional function spaces. The earliest Hilbert Spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer) and ergodic theory which forms the mathematical underpinning of the study of thermodynamics. John von Neumann coined the term Hilbert Space for the abstract concept underlying many of these diverse applications. The success of Hilbert Space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert Spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert Space theory.
- Michael A. Parker(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
’’ Mathematics and nature have separate but intertwined existences. As will become evident later, one must allow for the possibility that human observation (a physical process, not a mathematical one) can affect the state of the natural system. Linear algebra is the natural mathematical language of quantum mechanics. The quantum mechanical wave functions live in Hilbert Space essentially de fi ned as a vector space with an inner product. For this reason, the chapter starts with fi nite and in fi nite dimensional vector and Hilbert Spaces and uses the Dirac notation to help unify the concepts for these spaces. The chapter provides intuitive pictures to show how a function can be imagined as a vector in the Hilbert Space. It discusses inner products, norms, closure and completeness, and dual spaces. Fourier, Cosine, and Sine series appear as examples of expansions in complete orthonormal sets of functions. The Minkowski space provides an example of a pseudo-inner product, and provides examples on the use of the metric matrix, and the tensor notation found in physical applications. 2.1 VECTOR AND Hilbert SpaceS Linear algebra comprises the natural language of quantum theory. The present section provides direction and motivation for the upcoming topical areas by fi rst introducing the role of operators and vectors in the quantum theory and then de fi ning vector and Hilbert Spaces. The full view of the Hilbert Space and its connection with the quantum theory will unfold over the next several chapters. 2.1.1 M OTIVATION FOR L INEAR A LGEBRA IN Q UANTUM T HEORY In the case of quantum theory, it has become evident that the physical world can be represented by the ideas of linear algebra (more precisely, the Hilbert Space). The abstract linear algebra focuses on operators and vectors but not necessarily on the matrix form. Thinking back to elementary studies of mechanics and electromagnetics, it becomes apparent that one can apply vectors and 31
eBook - PDF- Rodney A. Kennedy, Parastoo Sadeghi(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
The generalization of Euclidean space in the field of functional analysis is associated with a different abstraction by Hilbert and other researchers in the 1900s and called Hilbert Space. 4 Introduction optimal control, filtering and equalization, signal processing on 2-sphere, Shan- non information theory, communication theory, linear and nonlinear stability theory, and many more. 1.1.3 Broadbrush structure Notion (cocktail party definition). A Hilbert Space is a complete inner product space. This is fine, except we are yet to define precisely what we mean by space, inner product and the adjective complete. But at a cocktail party where the objective is to impress strangers, particularly of the opposite sex, then it doesn’t matter. Notion (broad definition). The term “vector” is ingrained in early mathematical education as an ordered finite list of scalars, but in Hilbert Space it is a more general notion. We will alternatively use the term point in lieu of vector when the situation is not ambiguous. So when working in Hilbert Spaces the word vector might represent a conventional vector, a sequence or a function (and even more general objects). There are four key parts to a Hilbert Space: vector space, norm, inner product and completeness. We can hear the minimalists screaming already. 1 To have a degree of comfort with Hilbert Space is to have a clear notion of what these four things really mean and we will shortly move in the direction to address any deficiency. For the moment we are only interested in knowing what these mean in a general, possibly vague, way. Vector space Vector spaces should be familiar and align with the notions developed when dealing with the arithmetic of conventional vectors.
eBook - PDF- Eduard Prugovecki(Author)
- 1982(Publication Date)
- Academic Press(Publisher)
In quantum mechanics we are concerned primarily with separable complex Hilbert Spaces. We shall agree that in the future whenever we refer to a space as a Hilbert Space we mean a complex Hilbert Space, except if otherwise stated. Every subspace of a separable Euclidean space is a separable Euclidean space. of a Euclidean space d is also a Euclidean space is easy to check (see Exercise 2.6). In order to establish Theorem 4.2. Proof. The fact that a subspace 4. Hilbert Space 33 the separability of 8,, construct a countable subset S = {g,, , g,, , g,, , g,, ,...} of 8, in the following way. Let R = {fi , f 2 ,...I be a countable everywhere dense subset of 8; there has to be such a set because of the separability of &. Let g,,, denote a vector of 8, satisfying ( 1 g, - f, 1 1 < l/m, in case there is at least one such vector, or the zero vector in case there is no vector of 8, in the l/m neighborhood of fn . and m > 0 we can find an fn E R such that 1 1 h - fn 1 1 < l / m . Thus, by the above rule of constructing S we certainly have g, # 0 and therefore The set S is everywhere dense in 8 ' because for any given h E II h - gmn II d II h - fn II -t- Ilfn -grim II < 2im. This proves that S is everywhere dense in 8, . Q.E.D. 4.3. 1 ' SPACES AS EXAMPLES OF SEPARABLE Hilbert SpaceS As an important example of an infinite-dimensional separable Hilbert Theorem 4.3. The set Z2(m) of all one-column complex matrices 01 space we give the space Z2(co), which is basic in matrix mechanics. with a countable number of elements for which (4.5) becomes a separable Hilbert Space, denoted by Z2( a), if the vector opera- tions are defined by (4.7) and the inner product by (4-8) W ( a I P> = c a,*b, * k=l 34 I. Basic Ideas of Hilbert Space Theory Pyoof. The operation (4.7) maps C1 x Z2(m) into Z2(co) because xr=l I auk l 2 = I a l 2 xg1 I a k l 2 < +00 if (4.5) is satisfied.
eBook - PDF- Ronald. G. Douglas(Author)
- 1972(Publication Date)
- Academic Press(Publisher)
3.10 Proposition (Pythagorean Theorem) If {fl,fi,. ..&} is an orthogonal subset of the inner product space 9, then Proof Computing, we have 3.11 space 9, then Proposition (Parallelogram Law) Iff and g are in the inner product llf+g112 + Ilf-gll’ = 211fl12 + 211gl12 Proof Expand the left-hand side in terms of inner products. As in the case of normed linear spaces the deepest results are valid only if the space is complete in the metric induced by the norm. 3.12 Definition A Hilbert Space is a complex linear space which is com- plete in the metric induced by the norm. In particular, a Hilbert Space is a Banach space. 3.13 Examples We now consider some examples of Hilbert Spaces. For n a positive integer let @“ denote the collection of complex ordered n-tuples {x : x = (xI,xz, ..., x,), xi E C}. Then C“ is a complex linear space for the coordinate-wise operations. Define the inner product (,) on @” such that (x, y) = C;= xi ji. The properties of an inner product are easily verified and the associated norm is the usual Euclidean norm I/x/I2 = (C;= I 1.~~1’)”. To verify completeness suppose { x ’ ’ } ~ = ~ is a Cauchy sequence in C”. Then Examples of Hilbert Spaces: C, 1 2 , L2, and Ha 67 since Jxjl'-xim) < I ( x ~ -x ' J ~ ~ , it follows that {x:}?=~ is a Cauchy sequence in @ for I < i < n. If we set x = ( x , , x2,. . ., xk), where xi = limk-,a xjl', then x is in @ and limk-,a 2 = x in the norm of 63. Thus @ is a Hilbert Space. The space @ is the complex analog of real Euclidean n-space. We show later in this chapter, in a sense to be made precise, that the @'s are the only finite-dimensional Hilbert Spaces. 3.14 We next consider the union of the V's. Let 8 be the collection of complex functions on Z+ which take only finitely many nonzero values. With respect to pointwise addition and scalar multiplication, Y is a complex linear space. Moreover, (f, g) = x.,=of(n)s(n) defines an inner product on 9, where the sum converges, since all but finitely many terms are zero.
eBook - PDF- Eduard Prugovecki(Author)
- 2003(Publication Date)
- Academic Press(Publisher)
In quantum mechanics we are concerned primarily with separable complex Hilbert Spaces. We shall agree that in the future whenever we refer to a space as a Hilbert Space we mean a complex Hilbert Space, except if otherwise stated. Every subspace of a separable Euclidean space is a separable Euclidean space. The fact that a subspace 8‘’ of a Euclidean space € is also a Euclidean space is easy to check (see Exercise 2.6). In order to establish Theorem 4.2. Proof. 4. Hilbert Space 33 the separability of construct a countable subset S = {g,, , g,, , g,, , g,, ,...} of 8, in the following way. Let R = {f, , fi ,...} be a countable everywhere dense subset of 8; there has to be such a set because of the separability of &. Let g , , denote a vector of 8, satisfying ( 1 g, - f, 1 1 < l/m, in case there is at least one such vector, or the zero vector in case there is no vector of & , in the l/m neighborhood off, . and m > 0 we can find an f, E R such that ( 1 h - f, I( < l/m. Thus, by the above rule of constructing S we certainly have g , , # 0 and therefore The set S is everywhere dense in & , because for any given h E II h - gmn II < I/ h - fn I/ + II fn - gnm II < 2/m. This proves that S is everywhere dense in Q.E.D. 4.3. Z2 SPACES AS EXAMPLES OF SEPARABLE Hilbert SpaceS space we give the space Z2( m), which is basic in matrix mechanics. with a countable number of elements As an important example of an infinite-dimensional separable Hilbert Theorem 4.3. The set Z2( 00) of all one-column complex matrices 01 for which (4.5) m becomes a separable Hilbert Space, denoted by Z2( GO), if the vector opera- tions are defined by (4.6) . + B = ( a 2 t b 2 , ; ') aa = f;:), a E C', (4.7) and the inner product by (4.8) m ( a 1 p> = 2 ak*bk k=l 34 I. Basic Ideas of Hilbert Space Theory Proof. The operation (4.7) maps C1 x Z2(co) into Z2(co) because Cgl I auk j 2 = I a j 2 C& I ak l2 < + co if (4.5) is satisfied.
eBook - PDF- Michael T. Vaughn(Author)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
The standard textbook by Byron and Fuller cited in Chapter 2 emphasizes linear vector spaces. There are many modern books on quantum mechanics that discuss the essential connection between linear vector spaces and quantum mechanics. Two good introductory books are David J. Griffiths, Introduction to Quantum Mechanics (2nd edition), Prentice-Hall (2004). Ramamurti Shankar, Principles of Quantum Mechanics (2nd edition), Springer (2005) Slightly more advanced but still introductory is Eugen Merzbacher, Quantum Mechanics (3rd edition), Wiley (1997). Two classics that strongly reflect the original viewpoints of their authors are Paul A. M. Dirac, The Principles of Quantum Mechanics ( 4 th edition), Clarendon Press, Oxford (1958) John von Neumann, Mathematical Foundations of Quantum Mechanics , Princeton University Press (1955). Dirac’s work describes quantum mechanics as the mathematics flows from his own physi-cal insight, while von Neumann presents an axiomatic formulation based on his deep under-standing of Hilbert Space theory. Both are important works for the student of the historical development of the quantum theory. Problems 345 Problems 1. Let { φ n } ( n = 1 , 2 , . . . ) be a complete orthonormal system in the (infinite-dimensional) Hilbert Space H . Consider the operators U k defined by U k φ n = φ n + k ( k = 1 , 2 , . . . ). (i) Give an explicit form for U † k . (ii) Find the eigenvalues and eigenvectors of U k and U † k . (iii) Discuss the convergence of the sequences { U k } , { U † k } , { U † k U k } , { U k U † k } . 2. Let { φ n } ( n = 1 , 2 , . . . ) be a complete orthonormal system in H , and define the linear operator T by T φ n ≡ nφ n +1 ( n = 1 , 2 , . . . ) (i) What is the domain of T ? (ii) How does T † act on { φ 1 , φ 2 , . . . } ? What is the domain of T † ? (iii) Find the eigenvalues and eigenvectors of T . (iv) Find the eigenvalues and eigenvectors of T † . 3.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
