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

Physical Review

Communication Relations of Quantum Mechanics

Joseph V. Lepore

Published Jul 15, 1960• 6 citations
Publisher

Abstract

The mathematical and physical meaning of the commutation relations of nonrelativistic quantum mechanics is discussed in terms of the representation of translations, Galilean transformations, and rotations of the coordinate system by unitary transformations acting on the unitary vector space of quantum states.
DOI
10.1103/physrev.119.821

Related papers

Invariance Principles in Classical and Quantum Mechanics

Bernard Chern, Arnold Tubis

A simple, unified discussion of invariance principles in nonrelativistic classical and quantum mechanics is presented. In particular, space-translation, space-rotation, space-inversion, Galilean, and time-reversal invariance are discussed from the point of view of possible physical motions. In addition, there is a discussion of the classical connection between space inversion, time reversal, and what may be considered a type of “charge conjugation.”

American Journal of Physics•Mar 1, 1967•6 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

Quantum Mechanics

Jørgen Rammer

This chapter introduces the quantum mechanics. It explores the formulation in terms of the operator calculus. The chapter shows how nonrelativistic quantum kinematics emerges from the general theory. It discusses how physical quantities are related to symmetries through unitary transformations, and consider translations. The chapter also introduces the statistical operator and the density matrix. It introduces the consistent history interpretation of quantum mechanics. The true core of quantum mechanics is the rule for assigning the amplitude to an event that can be effected in indistinguishable ways. The chapter examines the presentation of Feynman's space-time approach since it offers the most intuitive formulation of quantum mechanics. A physical quantity is said to be conserved if it for any state of the system stays constant in time. In quantum mechanics this therefore means that its probability distribution in any state is independent of time.

CRC Press eBooks•May 4, 2018•0 citations

Understanding of the Representation and Its Transformation In Quantum Mechanics

Yao Jiu-min

Elementary theory of representation is introduced firstly,including primary meaning,expressions of a state and an operator in various representations,as well as the transformation from one representation to another.Then,representation and its transformation can be understood from its own instinct physical viewpoint through analogying with the counterpart in geometric coordinates and linear algebra mathematically,

Journal of Tangshan Teachers College•Jan 1, 2009•0 citations

Concluding and Proving the Commutation Relation of Operators often Used in Quantum Mechanics

Wan Meng

In order to help physical students really understanding the commutation relation of operators,we analyze,conclude and summarize the commutation relations of operators often used in quantum mechanics,which provide some references for teaching of quantum mechanics course as well.

Journal of Zunyi Normal College•Jan 1, 2011•0 citations

Weyl quantization

Giampiero V. M. Esposito, Giuseppe Marmo, George Sudarshan

The commutator of position and momentum operators is first considered in the coordinate and momentum representations, and then in an abstract Hilbert space. The reader is then introduced to canonical operators with the associated canonical quantization of commutation rules, which consists of maps defined on a symplectic space and taking values in the set of unitary operators on a Hilbert space. If the symplectic space is finite-dimensional, all of its irreducible representations are unitarily equivalent. The Weyl exponentiated form of the commutation relations is described and investigated. One then arrives at the definition of Weyl systems, and it is shown how to recover from them the formulation in terms of self-adjoint operators which satisfy a generalized form of commutation relations. Further topics are the Heisenberg representation for temporal evolution and the generalized uncertainty relations. Unitary operators associated with symplectic linear transformations are then considered, and, within this framework, translations, rotations and the harmonic oscillator are considered. The Weyl programme is then reassessed. Lastly, the basic postulates of modern quantum theory are eventually studied: the probabilistic nature of the predictions of results of measurements; the correspondence between observables and self-adjoint operators on Hilbert spaces; the expected result of measuring an observable quantity; and the evolution of the state vector in systems which are not affected by any external influence. The chapter ends with an outline of rigged Hilbert space formalism and its applications. The commutator in wave mechanics In the first part of this book we studied wave mechanics within the framework of the Schrödinger equation.

Cambridge University Press eBooks•Mar 11, 2004•0 citations
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 introduces the quantum mechanics. It explores the formulation in terms of the operator calculus. The chapter shows how nonrelativistic quantum kinematics emerges from the general theory. It discusses how physical quantities are related to symmetries through unitary transformations, and consider translations. The chapter also introduces the statistical operator and the density matrix. It introduces the consistent history interpretation of quantum mechanics. The true core of quantum mechanics is the rule for assigning the amplitude to an event that can be effected in indistinguishable ways. The chapter examines the presentation of Feynman's space-time approach since it offers the most intuitive formulation of quantum mechanics. A physical quantity is said to be conserved if it for any state of the system stays constant in time. In quantum mechanics this therefore means that its probability distribution in any state is independent of time.
The commutator of position and momentum operators is first considered in the coordinate and momentum representations, and then in an abstract Hilbert space. The reader is then introduced to canonical operators with the associated canonical quantization of commutation rules, which consists of maps defined on a symplectic space and taking values in the set of unitary operators on a Hilbert space. If the symplectic space is finite-dimensional, all of its irreducible representations are unitarily equivalent. The Weyl exponentiated form of the commutation relations is described and investigated. One then arrives at the definition of Weyl systems, and it is shown how to recover from them the formulation in terms of self-adjoint operators which satisfy a generalized form of commutation relations. Further topics are the Heisenberg representation for temporal evolution and the generalized uncertainty relations. Unitary operators associated with symplectic linear transformations are then considered, and, within this framework, translations, rotations and the harmonic oscillator are considered. The Weyl programme is then reassessed. Lastly, the basic postulates of modern quantum theory are eventually studied: the probabilistic nature of the predictions of results of measurements; the correspondence between observables and self-adjoint operators on Hilbert spaces; the expected result of measuring an observable quantity; and the evolution of the state vector in systems which are not affected by any external influence. The chapter ends with an outline of rigged Hilbert space formalism and its applications. The commutator in wave mechanics In the first part of this book we studied wave mechanics within the framework of the Schrödinger equation.