Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Oxford University Press eBooks

Operators

Gary E. Bowman

Published Nov 1, 2007• 0 citations
Publisher

Abstract

This chapter develops a basic working knowledge of operators — objects central to quantum mechanics. The fact that quantum operators are linear determines how they may be manipulated, so linearity forms the chapter's first topic. The adjoint of an operator is then introduced. This provides the basis for Hermitian or self-adjoint operators, which are discussed for both discrete eigenvalue systems and wavefunctions. Projection operators, which ‘project out’ a certain part of a quantum state, are then introduced. These form the basis for the identity operator — a sum over projection operators — which changes the representation in which a state is expressed without changing the state itself. Finally, unitary operators are briefly discussed, including the defining unitarity condition, the fact that such operators preserve inner products, and their physical meaning as effecting transformations in space and time. Unitary operators are discussed in greater detail in Appendix D.

DOI
10.1093/acprof:oso/9780199228928.003.0005

Related papers

Operators

Belal Ehsan Baaquie

Operators represent physically observable quantities, as discussed in Section 2.5. The structure and property of operators depend on the nature of the degree of freedom; operators act on the state space and in particular on the state vector of a given degree of freedom. The significance of operators in the interpretation of quantum mechanics has been discussed in Baaquie (2013e). The operators discussed in this chapter are mostly based on the continuous degree of freedom, which is analyzed in Section 3.1. Hermitian operators represent physically observable properties of a degree of freedom and their mathematical properties are defined in Section 3.3. The coordinate and momentum operators are the leading exemplar of a pair of noncommuting Hermitian operators and these are studied in some detail in Section 3.4. The Weyl operators yield, as in Section 3.5, a finite-dimensional example of the shift and scaling operators; Section 3.8 provides a unitary representation of the coordinate and momentum operators. The term self-adjoint operator is used for Hermitian operators when there is a need to emphasize the importance of the domain of the Hilbert space on which the operators act – a topic not usually discussed in most books on quantum mechanics. Sections 3.10 and 3.11 discuss the concept of self-adjoint operators, in particular the crucial role played by the domain for realizing the property of self-adjointness. It is shown in Section 3.12 how the requirement of self-adjointness yields a non-trivial extension of Hamiltonians that include singular interactions.

Cambridge University Press eBooks•Mar 27, 2014•0 citations

Observables as Operators

Tilak Sinha

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.

Chapman & Hall/CRC eBooks•Jun 18, 2021•0 citations

Mathematical Formulation of Quantum Mechanics

Rocco Schiavilla

The state space of a quantum system is defined; kets and bras are introduced, and their inner and outer products are defined; adjoint, hermitian , and unitary operators are introduced; representations of states and operators on discrete and/or continuous bases are discussed; the properties of commuting hermitian operators are examined; tensor products are defined.

Cambridge University Press eBooks•Dec 5, 2024•0 citations

Operators in quantum mechanics

Victor M. Galitski, Boris Mikhailovich Karnakov, Vladimir I Kogan +1 more

Abstract Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by Hermitian linear operators that act on state vectors that belong to a Hilbert vector space. This chapter analyzes problems that deal with the basic concepts of the theory of linear operators; eigenfunctions, eigenvalues, mean values; the projection operators; quantum-mechanical representations of operators and wave-functions; and unitary operators.

•Feb 28, 2013•2 citations

Elementary introduction to Lie algebras and operator methods

Francisco Marcelo Fernández, Eduardo A. Castro

This chapter provides a simple general introduction to some basic mathematics necessary for a subsequent discussion of Lie algebras, operator methods, and their applications to theoretical physics and chemistry. It also provides some useful mathematical tools to derive important theoretical results and expressions in a simple and elegant way. In quantum mechanics the state of a system is represented by an element of a particular vector space, in which one defines an appropriate scalar or inner product, known in mathematics as Hilbert space. Hermitian operators with a complete set of eigenvectors belonging to that state space represent physical observables. Moreover, in certain cases it proves useful to work with vector spaces formed by linear operators. Hermitian operators are particular cases of linear operators and physical observables are particular cases of Hermitian operators. In many physical applications of operator methods one needs to reorder products of exponential operators or to rewrite one such operator as a product of simpler ones.

CRC Press eBooks•Jan 16, 2020•0 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 eBooks•Oct 3, 2018•0 citations
This chapter develops a basic working knowledge of operators — objects central to quantum mechanics. The fact that quantum operators are linear determines how they may be manipulated, so linearity forms the chapter's first topic. The adjoint of an operator is then introduced. This provides the basis for Hermitian or self-adjoint operators, which are discussed for both discrete eigenvalue systems and wavefunctions. Projection operators, which ‘project out’ a certain part of a quantum state, are then introduced. These form the basis for the identity operator — a sum over projection operators — which changes the representation in which a state is expressed without changing the state itself. Finally, unitary operators are briefly discussed, including the defining unitarity condition, the fact that such operators preserve inner products, and their physical meaning as effecting transformations in space and time. Unitary operators are discussed in greater detail in Appendix D.
Operators represent physically observable quantities, as discussed in Section 2.5. The structure and property of operators depend on the nature of the degree of freedom; operators act on the state space and in particular on the state vector of a given degree of freedom. The significance of operators in the interpretation of quantum mechanics has been discussed in Baaquie (2013e). The operators discussed in this chapter are mostly based on the continuous degree of freedom, which is analyzed in Section 3.1. Hermitian operators represent physically observable properties of a degree of freedom and their mathematical properties are defined in Section 3.3. The coordinate and momentum operators are the leading exemplar of a pair of noncommuting Hermitian operators and these are studied in some detail in Section 3.4. The Weyl operators yield, as in Section 3.5, a finite-dimensional example of the shift and scaling operators; Section 3.8 provides a unitary representation of the coordinate and momentum operators. The term self-adjoint operator is used for Hermitian operators when there is a need to emphasize the importance of the domain of the Hilbert space on which the operators act – a topic not usually discussed in most books on quantum mechanics. Sections 3.10 and 3.11 discuss the concept of self-adjoint operators, in particular the crucial role played by the domain for realizing the property of self-adjointness. It is shown in Section 3.12 how the requirement of self-adjointness yields a non-trivial extension of Hamiltonians that include singular interactions.
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.
Abstract Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by Hermitian linear operators that act on state vectors that belong to a Hilbert vector space. This chapter analyzes problems that deal with the basic concepts of the theory of linear operators; eigenfunctions, eigenvalues, mean values; the projection operators; quantum-mechanical representations of operators and wave-functions; and unitary operators.
This chapter provides a simple general introduction to some basic mathematics necessary for a subsequent discussion of Lie algebras, operator methods, and their applications to theoretical physics and chemistry. It also provides some useful mathematical tools to derive important theoretical results and expressions in a simple and elegant way. In quantum mechanics the state of a system is represented by an element of a particular vector space, in which one defines an appropriate scalar or inner product, known in mathematics as Hilbert space. Hermitian operators with a complete set of eigenvectors belonging to that state space represent physical observables. Moreover, in certain cases it proves useful to work with vector spaces formed by linear operators. Hermitian operators are particular cases of linear operators and physical observables are particular cases of Hermitian operators. In many physical applications of operator methods one needs to reorder products of exponential operators or to rewrite one such operator as a product of simpler ones.
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.