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/0104094 · Physical Review A

Rigid rotor in phase space

Slobodan Danko Bosanac

Published Sep 12, 2001• 0 citations
arXivHTMLPDF

Abstract

Angular momentum is an important concept in physics, and its phase space properties are important in various applications. In this work a phase space analysis of angular momentum is made starting from its classical definition, and its quantum properties are obtained by imposing the uncertainty principle. The kinetic energy operator is derived, but its part have a different interpretation from that in the standard treatment. The rigid rotor is discussed and its phase space representation is shown. The true rigid rotor is defined and its phase space properties are also discussed.

DOI
10.1103/physreva.64.042708

Related papers

Angular Momentum and Rotational Motion in Three Dimensions

Jochen Autschbach

Abstract This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.

Oxford University Press eBooks•Dec 21, 2020•0 citations

Uncertainty principle for angular position and angular momentum

Sonja Franke‐Arnold, Stephen Mark Barnett, Eric Yao +3 more

The uncertainty principle places fundamental limits on the accuracy with which we are able to measure the values of different physical quantities (Heisenberg 1949 The Physical Principles of the Quantum Theory (New York: Dover); Robertson 1929 Phys. Rev. 34 127). This has profound effects not only on the microscopic but also on the macroscopic level of physical systems. The most familiar form of the uncertainty principle relates the uncertainties in position and linear momentum. Other manifestations include those relating uncertainty in energy to uncertainty in time duration, phase of an electromagnetic field to photon number and angular position to angular momentum (Vaccaro and Pegg 1990 J. Mod. Opt. 37 17; Barnett and Pegg 1990 Phys. Rev. A 41 3427). In this paper, we report the first observation of the last of these uncertainty relations and derive the associated states that satisfy the equality in the uncertainty relation. We confirm the form of these states by detailed measurement of the angular momentum of a light beam after passage through an appropriate angular aperture. The angular uncertainty principle applies to all physical systems and is particularly important for systems with cylindrical symmetry.

New Journal of Physics•Aug 9, 2004•265 citations

Momentum Impulse in Classical and Quantum Mechanics

Аташов И. А., Tagayev Marat Baymuratovich

This article examines the concept of angular momentum in classical and quantum mechanics. The introduction defines angular momentum and notes its importance for physics. A comparison is made of approaches to angular momentum in classical and quantum physics, and the main differences are revealed. In conclusion, the fundamental importance of angular momentum for understanding the nature and the possibility of its further study is emphasized.

International Journal of Informatics and Data Science Research•Jun 28, 2024•0 citations

Representation of rotations, angular momentum and spin

K. Heyde, John L. Wood

The various representations of rotations in physical space and Hilbert space are developed in detail. This leads to an in-depth treatment of the representation of states of well-defined spin and angular momentum in quantum systems. The peculiarities of the physics of spin— systems (spinors) is outlined. The tensorial character of representations is implicit in the treatment. The Schwinger and Bargmann representations are introduced in some detail; and this leads to SU(2) coherent states (which are important for more advanced group representation theory).

Institute of Physics eBooks•Apr 1, 2020•0 citations

Rotation and Angular Momentum

Nandita Rudra

The angular momentum operator is closely related to rotation in space. The quantum mechanical definition of angular momentum differs from the classical definition r × p of angular momentum. This expression is not sufficiently general as it cannot yield spin angular momentum which does not have a classical counterpart without having any coordinate representation.

CRC Press eBooks•Aug 19, 2019•1 citation

Unifying uncertainties for rotorlike quantum systems

Ladislav Mišta, Matouš Mišta, Z. Hradil

The quantum rotor represents, after the harmonic oscillator, the next obvious quantum system to study the complementary pair of variables: the angular momentum and the unitary shift operator in angular momentum. Proper quantification of uncertainties and the incompatibility of these two operators are thus essential for applications of rotorlike quantum systems. While angular momentum uncertainty is characterized by variance, several uncertainty measures have been proposed for the shift operator, with dispersion the simplest example. We establish a hierarchy of those measures and corresponding uncertainty relations which are all perfectly or almost perfectly saturated by a tomographically complete set of von Mises states. Building on the interpretation of dispersion as the moment of inertia of the unit ring we then show that the other measures also possess the same mechanical interpretation. This unifying perspective allows us to express all measures as a particular instance of a single generic angular uncertainty measure. The importance of these measures is then highlighted by applying the simplest two of them to derive optimal simultaneous measurements of the angular momentum and the shift operator. Finally, we argue that the model of quantum rotor extends beyond its mechanical meaning with promising applications in the fields of singular optics, superconductive circuits with a Josephson junction, or optimal pulse shaping in the time-frequency domain. Our findings lay the groundwork for quantum-information and metrological applications of the quantum rotor and point to its interdisciplinary nature.

arXiv:2403.02498•Physical Review A•Sep 9, 2024•2 citations
Angular momentum is an important concept in physics, and its phase space properties are important in various applications. In this work a phase space analysis of angular momentum is made starting from its classical definition, and its quantum properties are obtained by imposing the uncertainty principle. The kinetic energy operator is derived, but its part have a different interpretation from that in the standard treatment. The rigid rotor is discussed and its phase space representation is shown. The true rigid rotor is defined and its phase space properties are also discussed.
Abstract This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.
The uncertainty principle places fundamental limits on the accuracy with which we are able to measure the values of different physical quantities (Heisenberg 1949 The Physical Principles of the Quantum Theory (New York: Dover); Robertson 1929 Phys. Rev. 34 127). This has profound effects not only on the microscopic but also on the macroscopic level of physical systems. The most familiar form of the uncertainty principle relates the uncertainties in position and linear momentum. Other manifestations include those relating uncertainty in energy to uncertainty in time duration, phase of an electromagnetic field to photon number and angular position to angular momentum (Vaccaro and Pegg 1990 J. Mod. Opt. 37 17; Barnett and Pegg 1990 Phys. Rev. A 41 3427). In this paper, we report the first observation of the last of these uncertainty relations and derive the associated states that satisfy the equality in the uncertainty relation. We confirm the form of these states by detailed measurement of the angular momentum of a light beam after passage through an appropriate angular aperture. The angular uncertainty principle applies to all physical systems and is particularly important for systems with cylindrical symmetry.
The various representations of rotations in physical space and Hilbert space are developed in detail. This leads to an in-depth treatment of the representation of states of well-defined spin and angular momentum in quantum systems. The peculiarities of the physics of spin— systems (spinors) is outlined. The tensorial character of representations is implicit in the treatment. The Schwinger and Bargmann representations are introduced in some detail; and this leads to SU(2) coherent states (which are important for more advanced group representation theory).
The quantum rotor represents, after the harmonic oscillator, the next obvious quantum system to study the complementary pair of variables: the angular momentum and the unitary shift operator in angular momentum. Proper quantification of uncertainties and the incompatibility of these two operators are thus essential for applications of rotorlike quantum systems. While angular momentum uncertainty is characterized by variance, several uncertainty measures have been proposed for the shift operator, with dispersion the simplest example. We establish a hierarchy of those measures and corresponding uncertainty relations which are all perfectly or almost perfectly saturated by a tomographically complete set of von Mises states. Building on the interpretation of dispersion as the moment of inertia of the unit ring we then show that the other measures also possess the same mechanical interpretation. This unifying perspective allows us to express all measures as a particular instance of a single generic angular uncertainty measure. The importance of these measures is then highlighted by applying the simplest two of them to derive optimal simultaneous measurements of the angular momentum and the shift operator. Finally, we argue that the model of quantum rotor extends beyond its mechanical meaning with promising applications in the fields of singular optics, superconductive circuits with a Josephson junction, or optimal pulse shaping in the time-frequency domain. Our findings lay the groundwork for quantum-information and metrological applications of the quantum rotor and point to its interdisciplinary nature.