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

arXiv:quant-ph/0509171 · European Journal of Physics

The ubiquitous XP commutator

Alberto C de la Torre

Published Jan 3, 2006• 0 citations
arXivHTMLPDF

Abstract

Position and momentum of a particle can take any value in a continuous spectrum; these values are independent but their indeterminacies are correlated; momentum and position are mutually the generators of the transformations in each other. It is shown in a concise way, how all these features arise solely from their commutation relation $[X,P]=i\hbar$. The article is complete and self contained, adequate for didactic use.

DOI
10.1088/0143-0807/27/2/005

Related papers

The ubiquitous XP commutator

Alberto C. De La Torre

Position and momentum of a particle can take any value in a continuous spectrum; these values are independent but their indeterminacies are correlated; momentum and position are mutually the generators of the transformations in each other. It is shown in a concise way, how all these features arise solely from their commutation relation $[X,P]=i\\hbar$. The article is complete and self contained, adequate for didactic use.

•Jan 1, 2005•3 citations

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

A Concise Study about Fourier Transform and Principles of Quantum Mechanics

Do Tan Si

Based on the hypothesis that the position-representation of a physical state <x|\(\alpha\)> is the Fourier transform of its momentum-representation <P|\(\alpha\)> and that the time-representation <t|\(\alpha\)> is the inverse Fourier transform of its energy-representation <E|\(\alpha\)>, we are able to find the Planck relation E = hv, the de Broglie relation P = \(\frac{h}{\lambda}\). Afterward, utilizing the Dirac delta function \(\delta\)(x) and the property FTxf(x) = -i\(\hbar\)\(\partial\), pFTxf(x) = -i\(\hbar\)\(\partial\), pF(p) we obtain the links between operators in ordinary and Hilbert spaces leading to the Dirac fundamental commutation relation [\(\hat{p}, \hat{x}\)] = -i\(\hbar\)\(\hat{I}\), the Schrödinger equations, the Heisenberg uncertainty principle in quantum mechanics, the annihilation & creation of a photon from excitation & de-excitation of an atom according to Bohr.

Book Publisher International (a part of SCIENCEDOMAIN International)•Oct 31, 2022•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

Quantum mechanics of a free particle from properties of the Dirac delta function

Denys I. Bondar, Robert R. Lompay, Wing-Ki Liu

Based on the assumption that the probability density of finding a free particle is independent of position, we infer the form of the eigenfunction for the free particle, ⟨x|p⟩=exp(ipx/ℏ)/2πℏ. The canonical commutation relation between the momentum and position operators and the Ehrenfest theorem in the free particle case are derived solely from differentiation of the delta function and the form of ⟨x|p⟩.

arXiv:1007.4243•American Journal of Physics•Mar 14, 2011•6 citations

On position and momentum operators in the q-oscillator algebra

Won-Sang Chung, Anatoli U. Klimyk

The aim of this paper is to study the position and momentum operators in q-deformed oscillator algebras. The natural form of the position operator is Xp=qpN(a++a)qpN, where p is a real number. This operator is an operator representable by a Jacobi matrix. Using the theory of Jacobi matrices, the theory of classical moment problem and the theory of basic hypergeometric functions, it is shown that, depending on values of q and p, Xp can be unbounded symmetric operator [which has the deficiency indices (1,1) and, hence, is not self-adjoint, but has self-adjoint extensions], bounded self-adjoint operator with continuous simple spectrum or self-adjoint operator of trace class (therefore, with discrete spectrum with zero as the point of accumulation of eigenvalues). The connection of the q-deformed Heisenberg relation PX−qXP=1 for the position and momentum operators with a q-deformation of the quantum harmonic oscillator is also considered.

Journal of Mathematical Physics•Feb 1, 1996•33 citations
Position and momentum of a particle can take any value in a continuous spectrum; these values are independent but their indeterminacies are correlated; momentum and position are mutually the generators of the transformations in each other. It is shown in a concise way, how all these features arise solely from their commutation relation [X,P]=iℏ[X,P]=i\hbar[X,P]=iℏ. The article is complete and self contained, adequate for didactic use.
Position and momentum of a particle can take any value in a continuous spectrum; these values are independent but their indeterminacies are correlated; momentum and position are mutually the generators of the transformations in each other. It is shown in a concise way, how all these features arise solely from their commutation relation [X,P]=ihbar[X,P]=i\\hbar[X,P]=ihbar. The article is complete and self contained, adequate for didactic use.
Based on the hypothesis that the position-representation of a physical state <x|α\alphaα> is the Fourier transform of its momentum-representation <P|α\alphaα> and that the time-representation <t|α\alphaα> is the inverse Fourier transform of its energy-representation <E|α\alphaα>, we are able to find the Planck relation E = hv, the de Broglie relation P = hλ\frac{h}{\lambda}λh​. Afterward, utilizing the Dirac delta function δ\deltaδ(x) and the property FTxf(x) = -iℏ\hbarℏ∂\partial∂, pFTxf(x) = -iℏ\hbarℏ∂\partial∂, pF(p) we obtain the links between operators in ordinary and Hilbert spaces leading to the Dirac fundamental commutation relation [p^,x^\hat{p}, \hat{x}p^​,x^] = -iℏ\hbarℏI^\hat{I}I^, the Schrödinger equations, the Heisenberg uncertainty principle in quantum mechanics, the annihilation & creation of a photon from excitation & de-excitation of an atom according to Bohr.
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.
The aim of this paper is to study the position and momentum operators in q-deformed oscillator algebras. The natural form of the position operator is Xp=qpN(a++a)qpN, where p is a real number. This operator is an operator representable by a Jacobi matrix. Using the theory of Jacobi matrices, the theory of classical moment problem and the theory of basic hypergeometric functions, it is shown that, depending on values of q and p, Xp can be unbounded symmetric operator [which has the deficiency indices (1,1) and, hence, is not self-adjoint, but has self-adjoint extensions], bounded self-adjoint operator with continuous simple spectrum or self-adjoint operator of trace class (therefore, with discrete spectrum with zero as the point of accumulation of eigenvalues). The connection of the q-deformed Heisenberg relation PX−qXP=1 for the position and momentum operators with a q-deformation of the quantum harmonic oscillator is also considered.