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

Chapman & Hall/CRC eBooks

REPRESENTATION THEORY

Vishnu S. Mathur, Surendra Singh

Published Dec 12, 2008• 2 citations
Publisher

Abstract

The bra and ket vectors, as well as linear operators, are somewhat abstract mathematical quantities. Although the logic behind representing physical states and observables, respectively, by ket (or bra) vectors and self adjoint linear operators is perfect and, having developed the algebra for these quantities, we have been able to deduce therefrom some general results, identities and corollaries, yet these abstract quantities are not the ones in terms of which we can carry out numerical calculations and arrive at numbers to be compared with the experimental results. So we ask if it is possible to find more convenient mathematical quantities to represent ket vectors and linear operators that can be handled more conveniently. The answer is in the affirmative.

DOI
10.1201/b17176-5

Related papers

What are kets?

Yuri G. Gurevich, Andreas R. Blass

According to Dirac's bra-ket notation, in an inner-product space, the inner product $\langle x\,|\,y\rangle$ of vectors $x,y$ can be viewed as an application of the bra $\langle x|$ to the ket $|y\rangle$. Here $\langle x|$ is the linear functional $|y\rangle \mapsto \langle x\,|\,y\rangle$ and $|y\rangle$ is the vector $y$. But often -- though not always -- there are advantages in seeing $|y\rangle$ as the function $a \mapsto a\cdot y$ where $a$ ranges over the scalars. For example, the outer product $|y\rangle\langle x|$ becomes simply the composition $|y\rangle \circ \langle x|$. 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.

arXiv:2405.10055•May 16, 2024•0 citations

Rigorous bra-ket formalism and wave function operator for one particle quantum mechanics

Hervé Bergeron

Following previous works dedicated to the mathematical meaning of the “bra-ket” formalism [I. M. Gel’fand and G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. I; J. P. Antoine, J. Math. Phys. 10, 53 (1969); Yu. M. Berezanskii, Expansions of Self-adjoint Operators (American Mathematical Society, Providence, RI 1968); E. Prugovečki, J. Math. Phys. 14, 1410 (1973); J. P. Antoine and A. Grossmann, J. Funct. Anal. 23, 369 (1976); 23, 379 (1976)], we develop a new rigorous mathematical approach, based on an operator representation of bras and kets. This leads to a formalism very similar to second quantization. Well-defined operators associated with local observables can be exhibited, intimately related to previous works of E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space-Time (Reidel, Dordrecht, 1986)].

Journal of Mathematical Physics•Feb 1, 2006•3 citations

Antilinear operators in Dirac’s bra–ket notation

Antoine Royer

A notation is introduced which makes the manipulation of antilinear operators within Dirac’s bra–ket notation as efficient as that of linear operators.

American Journal of Physics•Aug 1, 1994•1 citation

Mathematical Background

F. S. Levin

The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.

Cambridge University Press eBooks•Dec 3, 2001•0 citations

Basic formalism

Bipin R. Desai

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.

Cambridge University Press eBooks•Dec 3, 2009•0 citations

Dirac Bra-Ket Notation

Z. Ficek

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.

•Mar 3, 2017•0 citations
The bra and ket vectors, as well as linear operators, are somewhat abstract mathematical quantities. Although the logic behind representing physical states and observables, respectively, by ket (or bra) vectors and self adjoint linear operators is perfect and, having developed the algebra for these quantities, we have been able to deduce therefrom some general results, identities and corollaries, yet these abstract quantities are not the ones in terms of which we can carry out numerical calculations and arrive at numbers to be compared with the experimental results. So we ask if it is possible to find more convenient mathematical quantities to represent ket vectors and linear operators that can be handled more conveniently. The answer is in the affirmative.
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.
Following previous works dedicated to the mathematical meaning of the “bra-ket” formalism [I. M. Gel’fand and G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. I; J. P. Antoine, J. Math. Phys. 10, 53 (1969); Yu. M. Berezanskii, Expansions of Self-adjoint Operators (American Mathematical Society, Providence, RI 1968); E. Prugovečki, J. Math. Phys. 14, 1410 (1973); J. P. Antoine and A. Grossmann, J. Funct. Anal. 23, 369 (1976); 23, 379 (1976)], we develop a new rigorous mathematical approach, based on an operator representation of bras and kets. This leads to a formalism very similar to second quantization. Well-defined operators associated with local observables can be exhibited, intimately related to previous works of E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space-Time (Reidel, Dordrecht, 1986)].
The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.
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.
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.