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

Self-adjoint momentum operator for a particle confined in a multi-dimensional cavity

A. Mariani, U.‐J. Wiese

Published Apr 1, 2024• 3 citations
Publisher

Abstract

Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.

DOI
10.1063/5.0178419

Related papers

Self-adjoint Momentum Operator for a Particle Confined in a Multi-Dimensional Cavity

A. Mariani, U.‐J. Wiese

Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.

arXiv:2309.07818•Sep 14, 2023•0 citations

Confined quantum systems and their limits

F. Belgiorno, Franco Gallone

In nonrelativistic quantum mechanics, the notion of the confined version of a system is proposed and discussed. Confined observables can turn out to be represented by just positive operator valued measures also when their nonconfined equivalents are represented by self-adjoint operators, and this nearly always happens for confined linear momentum observables. However, the self-adjoint operators of the nonconfined system can still be used to compute expected results and variances of confined observables. As a result, the precision with which some confined observables can be measured has limits which ultimately derive, through the Heisenberg commutation rules, from the covariance properties of nonconfined kinematical observables. Two conditions are proposed to characterize the approximation of a nonconfined system by a sequence of confined ones. A nonconfined system is deemed to be approximated by a sequence of confined systems if all probabilities determined by any observable of the nonconfined system with respect to nonconfined states can be approximated by confined counterparts and if the time evolution of any state of the nonconfined system can be approximated in any finite time interval by the time evolution of confined states. A rather general example of this situation is discussed.

Journal of Mathematical Physics•Feb 1, 2009•3 citations

A New Concept for the Momentum of a Quantum Mechanical Particle in a Box

M.H. Al-Hashimi, U.‐J. Wiese

For a particle in a box, the operator $- i \partial_x$ is not Hermitean. We provide an alternative construction of a momentum operator $p = p_R + i p_I$, which has a Hermitean component $p_R$ that can be extended to a self-adjoint operator, as well as an anti-Hermitean component $i p_I$. This leads to a description of momentum measurements performed on a particle that is strictly limited to the interior of a box.

arXiv:2012.09596•Dec 17, 2020•1 citation

Alternative momentum concept for a quantum mechanical particle in a box

M.H. Al-Hashimi, U.‐J. Wiese

For a particle in a box, the operator p = -i∂ x is not self-adjoint.We provide an alternative construction of a momentum operator pR + i pI , which has two self-adjoint components pR and pI .This leads to a description of momentum measurements performed on a particle that is strictly limited to the interior of a box of size L, which yields quantized momentum values π n/L with n ∈ Z.

Physical Review Research•Oct 26, 2021•11 citations

Bouncing Wave Packets, Ehrenfest Theorem, and Uncertainty Relation based upon a new Concept for the Momentum of a Particle in a Box

I. Albrecht, J. Herrmann, A. Mariani +2 more

For a particle in a box, the operator $-i\partial_x$ is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg-Robertson-Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it.

arXiv:2206.07531•Jun 15, 2022•1 citation

Bouncing wave packets, Ehrenfest theorem, and uncertainty relation based upon a new concept for the momentum of a particle in a box

I. Albrecht, Joachım Herrmann, A. Mariani +2 more

For a particle in a box, the operator −i∂x is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg–Robertson–Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it.

Annals of Physics•Mar 22, 2023•7 citations
Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.
Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.
In nonrelativistic quantum mechanics, the notion of the confined version of a system is proposed and discussed. Confined observables can turn out to be represented by just positive operator valued measures also when their nonconfined equivalents are represented by self-adjoint operators, and this nearly always happens for confined linear momentum observables. However, the self-adjoint operators of the nonconfined system can still be used to compute expected results and variances of confined observables. As a result, the precision with which some confined observables can be measured has limits which ultimately derive, through the Heisenberg commutation rules, from the covariance properties of nonconfined kinematical observables. Two conditions are proposed to characterize the approximation of a nonconfined system by a sequence of confined ones. A nonconfined system is deemed to be approximated by a sequence of confined systems if all probabilities determined by any observable of the nonconfined system with respect to nonconfined states can be approximated by confined counterparts and if the time evolution of any state of the nonconfined system can be approximated in any finite time interval by the time evolution of confined states. A rather general example of this situation is discussed.
For a particle in a box, the operator −i∂x- i \partial_x−i∂x​ is not Hermitean. We provide an alternative construction of a momentum operator p=pR+ipIp = p_R + i p_Ip=pR​+ipI​, which has a Hermitean component pRp_RpR​ that can be extended to a self-adjoint operator, as well as an anti-Hermitean component ipIi p_IipI​. This leads to a description of momentum measurements performed on a particle that is strictly limited to the interior of a box.
For a particle in a box, the operator −i∂x-i\partial_x−i∂x​ is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg-Robertson-Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it.
For a particle in a box, the operator −i∂x is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg–Robertson–Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it.