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

The commutation relations

Keith Hannabuss

Published Mar 20, 1997• 1 citation
Publisher

Abstract

Abstract Our results so far, in particular Section 3.4, strongly suggest that although it is possible to find formalisms in which either momentum or position are easy to handle, one cannot simultaneously manage both. The underlying reason for this is that the operators P and X do not commute.
DOI
10.1093/oso/9780198537946.003.0007

Related papers

An Alternative Commutation Relation Between Position and Momentum Operators of Massless Particles

H. Razmi

It seems that the problem of finding a suitable position operator for photon has been solved in a recently published work which is based on a new commutation relation between position and momentum operators of massless particles[1]. Although the authors of [1] have presented a classical reasoning in favour of the new commutator,here we are going to find this new commutator based on a quantum mechanical argument.

arXiv:quant-ph/9811071•Nov 25, 1998•0 citations

Introduction to Representations of the Canonical Commutation and Anticommutation Relations

Jan Dereziński

Since the early days of quantum mechanics it has been noted that the position operator x and the momentum operator D := −i∇ satisfy the following commutation relation:

•Jan 1, 2005•39 citations

Angular momentum operator commutator against position and Hamiltonian of a free particle

Bambang Supriadi, Trapsilo Prihandono, V Rizqiyah +3 more

Observables in Quantum mechanicsare represented by operators. Its measurements cannot be determined simultaneously and the results are only a possibility. This study uses the rules of the commutator in determining the relation between angular momentum to the position and free particle Hamiltonian. The commutator will be zero if the operator that is connected can be determined simultaneously, and it will produce value if cannot be determined simultaneously. This study focused on the commutator of the angular momentum operator on the position and Hamiltonian of free particles in Cartesian coordinates. The result of the commutator of angular momentum operator to the position was zero (commut) if there wasn't a component of the angular momentum that is equal to the position made by the commutation pair. While the results of the commutator angular momentum operator towards the free particle Hamiltonian indicated that angular momentum is the constant of motion.

Journal of Physics Conference Series•Apr 1, 2019•3 citations

On a discrete version of the position-momentum commutation relation

Nicolae Cotfas

arXiv:2509.22753•Jan 1, 2025•0 citations

Uncertainty Principles for Position- and Momentum-Like Operators

Lara Ortelli, Gael Silva

At the quantum scale, objects such as fundamental particles, atoms, and molecules are modeled as localized wavepackets rather than concrete objects. As such, some quantities that are well-defined at our human scale physics instead become uncertain. For instance, quantum objects do not have a precise position, and instead their position is understood probabilistically and we look to the expected, or average, value of the position. Another quirk of quantum mechanics is that quantities such as position and momentum become operators that act on functions, similar to matrices acting on vectors. As such, quantities that may have commuted classically may not at the quantum level. For instance, position (x) and momentum (p) commute in classical physics (xp = px), but they do not commute at the quantum level (xp ≠ px). For noncommuting operators A and B, the uncertainty principle qualitatively describes how well one can know A and B simultaneously: the more certain one is of the value of A, the less one knows the value B and vice versa. The standard example is that of position and momentum: the Heisenberg uncertainty principle states that the more certain one is of the position of a quantum object, the more uncertain one is about its momentum. The classic uncertainty principles between noncommuting operators are those of position and momentum, the z and x or y components of angular momentum, and number and phase operators. These operators are very rigid and do not provide easy generalization to more general operators. In this project, we used an infinite matrix representation to analyze the uncertainty principle underlying position and momentum which allows for broader generalization. The biggest component of this project was determining eigenvalues of infinite banded matrices which was aided by the use of the mathematical software Mathematica.

•Mar 27, 2026•0 citations

Chapter 10. Commutation Relations in Quantum Mechanics

Dennis Morris

De Gruyter eBooks•May 27, 2016•0 citations
Observables in Quantum mechanicsare represented by operators. Its measurements cannot be determined simultaneously and the results are only a possibility. This study uses the rules of the commutator in determining the relation between angular momentum to the position and free particle Hamiltonian. The commutator will be zero if the operator that is connected can be determined simultaneously, and it will produce value if cannot be determined simultaneously. This study focused on the commutator of the angular momentum operator on the position and Hamiltonian of free particles in Cartesian coordinates. The result of the commutator of angular momentum operator to the position was zero (commut) if there wasn't a component of the angular momentum that is equal to the position made by the commutation pair. While the results of the commutator angular momentum operator towards the free particle Hamiltonian indicated that angular momentum is the constant of motion.
At the quantum scale, objects such as fundamental particles, atoms, and molecules are modeled as localized wavepackets rather than concrete objects. As such, some quantities that are well-defined at our human scale physics instead become uncertain. For instance, quantum objects do not have a precise position, and instead their position is understood probabilistically and we look to the expected, or average, value of the position. Another quirk of quantum mechanics is that quantities such as position and momentum become operators that act on functions, similar to matrices acting on vectors. As such, quantities that may have commuted classically may not at the quantum level. For instance, position (x) and momentum (p) commute in classical physics (xp = px), but they do not commute at the quantum level (xp ≠ px). For noncommuting operators A and B, the uncertainty principle qualitatively describes how well one can know A and B simultaneously: the more certain one is of the value of A, the less one knows the value B and vice versa. The standard example is that of position and momentum: the Heisenberg uncertainty principle states that the more certain one is of the position of a quantum object, the more uncertain one is about its momentum. The classic uncertainty principles between noncommuting operators are those of position and momentum, the z and x or y components of angular momentum, and number and phase operators. These operators are very rigid and do not provide easy generalization to more general operators. In this project, we used an infinite matrix representation to analyze the uncertainty principle underlying position and momentum which allows for broader generalization. The biggest component of this project was determining eigenvalues of infinite banded matrices which was aided by the use of the mathematical software Mathematica.