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

Journal of Mathematical Physics

Canonically Conjugate Pairs, Uncertainty Relations, and Phase Operators

J. C. Garrison, Jack Wong

Published Aug 1, 1970• 155 citations
Publisher

Abstract

Apparent difficulties that prevent the definition of canonical conjugates for certain observables, e.g., the number operator, are eliminated by distinguishing between the Heisenberg and Weyl forms of the canonical commutation relations (CCR's). Examples are given for which the uncertainty principle does not follow from the CCR's. An operator F is constructed which is canonically conjugate, in the Heisenberg sense, to the number operator; and F is used to define a quantum time operator.

DOI
10.1063/1.1665388

Related papers

On Heisenberg's Uncertainty Principle and the CCR

Reinhard Honegger

Abstract Realizing the canonical commutation relations (CCR) [N, Θ] = - i as N = - i d/dϑ and Θ to be the multiplication by ϑ on the Hilbert space of square integrable functions on [0, 2π], in the physical literature there seems to be some contradictions concerning the Heisenberg uncertainty principle ⟨ΔN⟨ ⟨ΔΘ⟨ ≥ 1/4. The difficulties may be overcome by a rigorous mathematical analysis of the domain of state vectors, for which Heisenberg's inequality is valid. It is shown that the exponentials exp {i t N} and exp{i sΘ} satisfy some commutation relations, which are not the Weyl relations. Finally, the present work aims at a better understanding of the phase and number operators in non-Fock representations.

Zeitschrift für Naturforschung A•Mar 1, 1993•7 citations

When does "Time" become "Observable" ?

Tadashi Fujimoto

We investigate some spectral properties of time operators which are obtained through Canonical Commutation Relation (CCR) and Positive Operator Valued Measure (POVM) of quantum physics. In addition, we re-interpret the spectral properties of time operators from the standpoint of the Copenhagen Interpretation, especially, of W.Heisenberg.

arXiv:1403.1557•Mar 5, 2014•0 citations

Canonically conjugate pairs and phase operators

Kurt Schönhammer

For quantum mechanics on a lattice the position (``particle number'') operator and the quasimomentum (``phase'') operator obey canonical commutation relations (CCRs) only on a dense set of the Hilbert space. We compare exact numerical results for a particle in a linear and a quadratic potential on the lattice with the expectations, when the CCRs are assumed to be strictly obeyed. Only for sufficiently smooth eigenfunctions does this lead to reasonable results. In the long time limit the use of the CCRs can lead to a qualitatively wrong dynamics even if the initial state is in the dense set.

arXiv:quant-ph/0204139•Physical Review A•Jul 11, 2002•6 citations

Definition of Commutators via the Uncertainty Principle

Bryce Seligman DeWitt

A general demonstration is given of the fact that the quantization of any system, as expressed by the uncertainty principle, implies the quantization of all other systems to which it can be coupled. It is also shown that the precise form which the uncertainty principle takes is uniquely specified. That is to say, the uncertainty principle effectively defines the commutator of any pair of observables. The argument serves to detach completely from its canonical origins a previously given definition of the commutator (which is a generalization of one given by Peierls) applicable to systems with infinite-dimensional invariance groups, for which the identification of canonical variables is an inconvenience. The measurements of a single observable and of a pair of observables are analyzed according to the Bohr-Rosenfeld scheme.

Journal of Mathematical Physics•Jul 1, 1962•32 citations

Quantum-optical Phase and Canonical Conjugation

David Thomas Pegg, Joan A. Vaccaro, Stephen Mark Barnett

We use a new limiting procedure, developed to study quantum-optical phase, to examine canonically conjugate operators in general. We find that Dirac's assumption that photon number and phase should be canonically conjugate variables, similar to momentum and position, is essentially correct. The difficulties with Dirac's approach are shown to arise through use of a form of the canonical commutator which, although the only possible form in the usual infinite Hilbert space approach, is not sufficiently general to be used as a model for a number-phase commutator. The approach in this paper unifies the theory of conjugate operators, which include photon number and phase, angular momentum and angle, and momentum and position as particular cases. The usual position-momentum commutator is regained from a more generally applicable expression by means of a domain restriction which cannot be used for the phase-number commutator.

Journal of Modern Optics•Nov 1, 1990•71 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
Apparent difficulties that prevent the definition of canonical conjugates for certain observables, e.g., the number operator, are eliminated by distinguishing between the Heisenberg and Weyl forms of the canonical commutation relations (CCR's). Examples are given for which the uncertainty principle does not follow from the CCR's. An operator F is constructed which is canonically conjugate, in the Heisenberg sense, to the number operator; and F is used to define a quantum time operator.
Abstract Realizing the canonical commutation relations (CCR) [N, Θ] = - i as N = - i d/dϑ and Θ to be the multiplication by ϑ on the Hilbert space of square integrable functions on [0, 2π], in the physical literature there seems to be some contradictions concerning the Heisenberg uncertainty principle ⟨ΔN⟨ ⟨ΔΘ⟨ ≥ 1/4. The difficulties may be overcome by a rigorous mathematical analysis of the domain of state vectors, for which Heisenberg's inequality is valid. It is shown that the exponentials exp {i t N} and exp{i sΘ} satisfy some commutation relations, which are not the Weyl relations. Finally, the present work aims at a better understanding of the phase and number operators in non-Fock representations.
For quantum mechanics on a lattice the position (``particle number'') operator and the quasimomentum (``phase'') operator obey canonical commutation relations (CCRs) only on a dense set of the Hilbert space. We compare exact numerical results for a particle in a linear and a quadratic potential on the lattice with the expectations, when the CCRs are assumed to be strictly obeyed. Only for sufficiently smooth eigenfunctions does this lead to reasonable results. In the long time limit the use of the CCRs can lead to a qualitatively wrong dynamics even if the initial state is in the dense set.
A general demonstration is given of the fact that the quantization of any system, as expressed by the uncertainty principle, implies the quantization of all other systems to which it can be coupled. It is also shown that the precise form which the uncertainty principle takes is uniquely specified. That is to say, the uncertainty principle effectively defines the commutator of any pair of observables. The argument serves to detach completely from its canonical origins a previously given definition of the commutator (which is a generalization of one given by Peierls) applicable to systems with infinite-dimensional invariance groups, for which the identification of canonical variables is an inconvenience. The measurements of a single observable and of a pair of observables are analyzed according to the Bohr-Rosenfeld scheme.
We use a new limiting procedure, developed to study quantum-optical phase, to examine canonically conjugate operators in general. We find that Dirac's assumption that photon number and phase should be canonically conjugate variables, similar to momentum and position, is essentially correct. The difficulties with Dirac's approach are shown to arise through use of a form of the canonical commutator which, although the only possible form in the usual infinite Hilbert space approach, is not sufficiently general to be used as a model for a number-phase commutator. The approach in this paper unifies the theory of conjugate operators, which include photon number and phase, angular momentum and angle, and momentum and position as particular cases. The usual position-momentum commutator is regained from a more generally applicable expression by means of a domain restriction which cannot be used for the phase-number commutator.
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.