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

Physical Review

Complementarity in Angular-Momentum States

Bernard A. Lippmann

Published Sep 1, 1953• 0 citations
Publisher

Abstract

By introducing operator representations for angular-momentum states referred to arbitrary axes of quantization, two independent angular momenta, each referred to its own axis of quantization, may be combined quantum-mechanically. An interpretation in terms of complementarity shows that this method of adding angular momenta corresponds to a "causal" description, while the usual method (using only one axis of quantization) is a "geometric" or spacetime description. "Causal" angular-momentum states automatically avoid the sums over magnetic quantum numbers in cascade processes.

DOI
10.1103/physrev.91.1213

Related papers

Angular Momentum

Will Raven

Abstract In this chapter, we explore angular momentum, a key concept in quantum mechanics and atomic physics. We cover the quantization of angular momentum, the role of quantum numbers, and their impact on atomic states. We will learn about the different types of angular momentum, the states representing these types of angular momentum, and how angular momenta are added together in quantum mechanics. The chapter also examines the compatibility of different observables, such as the magnitude and components of angular momentum, and the impact of measurements on quantum states.

Springer eBooks•Oct 17, 2024•0 citations

Angular momentum in two dimensions

D. K. Sunko

It is shown by direct computation that angular momentum cannot be reduced to two dimensions by assuming lack of dependence on the z coordinate. Formally, this is because the independence of z breaks rotational invariance. As a physical example, it is shown that a gauge- and rotation-invariant description of planar electrons in a perpendicular magnetic field must have a three-dimensional form, even though all the terms refer to planar coordinates alone. The physical concepts underlying common formal manipulations with the quantum angular momentum operator are treated at a level directly useful to students encountering them thoroughly for the first time. The letter may also serve as a contextual introduction to advanced quantization issues, usually studied at graduate level.

European Journal of Physics•Apr 19, 2010•1 citation

Remarks on the Theory of Angular Momenta

O. Chavoya-Aceves

A rigorous application of the correspondence rules shows that the operator of the angular momentum of a quantum particle---corresponding to the classical magnitude $\mathbf{l}= m \mathbf{r} \wedge \mathbf{v}$---is given by $\mathbf{\hat{l}}=\mathbf{r}\wedge(-i\hbar\mathbf{\nabla} -\frac{e}{c}\mathbf{A})$ in the presence of an electromagnetic field. Thus, despite the general opinion on the corresponding rules of quantization, the eigenvalues of the angular momentum depend on the configuration of the electromagnetic field. The usual rules of commutation $[{\hat{l}}_i,{\hat{l}}_j]=i\hbarε_{ijk}{\hat{l}}_k$, that are at the foundation of the calculus of angular momentum and of the theory of \emph{spin}---and Bohm's example of the EPR argument---are not valid in the presence of an electromagnetic field. The expected value of the operator $\mathbf{\hat{l}}=-i\hbar\mathbf{r}\wedge\mathbf{\nabla}$ is not gauge invariant, it depends on the calibration of the electrodynamic potentials.

arXiv:quant-ph/0305049•May 9, 2003•2 citations

The exact meaning of the angular-momentum and spin operators in quantum mechanics

G. Coddens

The theory of angular momentum and spin in quantum mechanics seems to defy common-sense intuition.We render the theory intelligible again by pointing out that this apparent impenetrability merely stems from an {\em undue} parallel interpretation of the algebraic expressions for the angular-momentum and spin operators in the group representation theory of SO(3) and SU(2). E.g. the correct meaning of the quantumoperator ${\hat{L}}_{z} = {\hbar\over{\imath}}\,(x{\partial\over{\partial y}} - y{\partial\over{\partial x}} )$ is not that it is the operatorfor the $z$-component $L_{z}$ of the angular momentum ${\mathbf{L}}$, but rather the expression of the operator for the angular momentum ${\mathbf{L}}$ when it is aligned with the $z$-axis. Hence what we are used to note (erroneously) as ${\hat{L}}_{z}$ is not a scalar but a vector operator.The same applies{\em mutatis mutandis} for the spin operators. In the correct interpretation, the wholealgebraic formalism is just the group representation theory for the rotationsof three-dimensional Euclidean geometry. It is thus mere,elementary high-school mathematics (in a less usual, more technical guise) and as such totally exempt of any physics, let alone quantum mysteries.The change of interpretation has no impact on the algebraic results, such that they remain in agreement with experimental data and the algebra quantum mechanics remains correct. It is all only a matter of the correct geometrical meaning of the algebra.All these statements are proved within the framework of the group representation theory for SO(3) and SU(2) which is the basic tool used to describe rotational motion in quantum mechanics.

•Sep 11, 2021•0 citations

Addition of Angular Momenta

Walter Wilcox

Synopsis: General operator properties of angular momenta are recalled, and a physical motivation for adding them is given. The product and total angular momentum bases are defined. It is shown how to express one basis in terms of the other, and an explicit example of adding two spin 1/2 angular momenta is given. A further example consists of adding a general orbital angular momentum value to spin 1/2. These considerations are applied to perturbations in the hydrogen atom energy spectrum, where the correct choice for calculations is the total angular momentum basis.

CRC Press eBooks•Aug 23, 2019•1 citation

Dynamics of angular momentum coupling in an arbitrarily directed magnetic field

Michael Brieger

The choice of an axis of quantization automatically selects those species from a bulk which could become stationary in a field-free environment with related spatial orientations of their coupling angular momenta. But if this axis fails to coincide with the direction of a uniform external magnetic field, a field-imposed transverse torque forces these atoms' total angular momenta into steady nutation that cannot be abolished by coupling-related internal torques. Making any stationary state impossible for these atoms, it also causes the contributions to the energy expectation value to oscillate. This is the result of a rigorous solution to the time-dependent Schrödinger equation describing the coupling dynamics of two spins 1/2 in an external field. Demanding to solve a four-dimensional system of coupled linear differential equations of first order, the bulkiness of even this simplest case already brings the calculations to the verge of becoming infeasible analytically.

•Jan 1, 2000•0 citations
By introducing operator representations for angular-momentum states referred to arbitrary axes of quantization, two independent angular momenta, each referred to its own axis of quantization, may be combined quantum-mechanically. An interpretation in terms of complementarity shows that this method of adding angular momenta corresponds to a "causal" description, while the usual method (using only one axis of quantization) is a "geometric" or spacetime description. "Causal" angular-momentum states automatically avoid the sums over magnetic quantum numbers in cascade processes.
Abstract In this chapter, we explore angular momentum, a key concept in quantum mechanics and atomic physics. We cover the quantization of angular momentum, the role of quantum numbers, and their impact on atomic states. We will learn about the different types of angular momentum, the states representing these types of angular momentum, and how angular momenta are added together in quantum mechanics. The chapter also examines the compatibility of different observables, such as the magnitude and components of angular momentum, and the impact of measurements on quantum states.
It is shown by direct computation that angular momentum cannot be reduced to two dimensions by assuming lack of dependence on the z coordinate. Formally, this is because the independence of z breaks rotational invariance. As a physical example, it is shown that a gauge- and rotation-invariant description of planar electrons in a perpendicular magnetic field must have a three-dimensional form, even though all the terms refer to planar coordinates alone. The physical concepts underlying common formal manipulations with the quantum angular momentum operator are treated at a level directly useful to students encountering them thoroughly for the first time. The letter may also serve as a contextual introduction to advanced quantization issues, usually studied at graduate level.
A rigorous application of the correspondence rules shows that the operator of the angular momentum of a quantum particle---corresponding to the classical magnitude l=mr∧v\mathbf{l}= m \mathbf{r} \wedge \mathbf{v}l=mr∧v---is given by l^=r∧(−iℏ∇−ecA)\mathbf{\hat{l}}=\mathbf{r}\wedge(-i\hbar\mathbf{\nabla} -\frac{e}{c}\mathbf{A})l^=r∧(−iℏ∇−ce​A) in the presence of an electromagnetic field. Thus, despite the general opinion on the corresponding rules of quantization, the eigenvalues of the angular momentum depend on the configuration of the electromagnetic field. The usual rules of commutation [l^i,l^j]=iℏεijkl^k[{\hat{l}}_i,{\hat{l}}_j]=i\hbarε_{ijk}{\hat{l}}_k[l^i​,l^j​]=iℏεijk​l^k​, that are at the foundation of the calculus of angular momentum and of the theory of \emph{spin}---and Bohm's example of the EPR argument---are not valid in the presence of an electromagnetic field. The expected value of the operator l^=−iℏr∧∇\mathbf{\hat{l}}=-i\hbar\mathbf{r}\wedge\mathbf{\nabla}l^=−iℏr∧∇ is not gauge invariant, it depends on the calibration of the electrodynamic potentials.
The theory of angular momentum and spin in quantum mechanics seems to defy common-sense intuition.We render the theory intelligible again by pointing out that this apparent impenetrability merely stems from an {\em undue} parallel interpretation of the algebraic expressions for the angular-momentum and spin operators in the group representation theory of SO(3) and SU(2). E.g. the correct meaning of the quantumoperator L^z=ℏı (x∂∂y−y∂∂x){\hat{L}}_{z} = {\hbar\over{\imath}}\,(x{\partial\over{\partial y}} - y{\partial\over{\partial x}} )L^z​=ℏ​(x∂y∂​−y∂x∂​) is not that it is the operatorfor the zzz-component LzL_{z}Lz​ of the angular momentum L{\mathbf{L}}L, but rather the expression of the operator for the angular momentum L{\mathbf{L}}L when it is aligned with the zzz-axis. Hence what we are used to note (erroneously) as L^z{\hat{L}}_{z}L^z​ is not a scalar but a vector operator.The same applies{\em mutatis mutandis} for the spin operators. In the correct interpretation, the wholealgebraic formalism is just the group representation theory for the rotationsof three-dimensional Euclidean geometry. It is thus mere,elementary high-school mathematics (in a less usual, more technical guise) and as such totally exempt of any physics, let alone quantum mysteries.The change of interpretation has no impact on the algebraic results, such that they remain in agreement with experimental data and the algebra quantum mechanics remains correct. It is all only a matter of the correct geometrical meaning of the algebra.All these statements are proved within the framework of the group representation theory for SO(3) and SU(2) which is the basic tool used to describe rotational motion in quantum mechanics.
Synopsis: General operator properties of angular momenta are recalled, and a physical motivation for adding them is given. The product and total angular momentum bases are defined. It is shown how to express one basis in terms of the other, and an explicit example of adding two spin 1/2 angular momenta is given. A further example consists of adding a general orbital angular momentum value to spin 1/2. These considerations are applied to perturbations in the hydrogen atom energy spectrum, where the correct choice for calculations is the total angular momentum basis.
The choice of an axis of quantization automatically selects those species from a bulk which could become stationary in a field-free environment with related spatial orientations of their coupling angular momenta. But if this axis fails to coincide with the direction of a uniform external magnetic field, a field-imposed transverse torque forces these atoms' total angular momenta into steady nutation that cannot be abolished by coupling-related internal torques. Making any stationary state impossible for these atoms, it also causes the contributions to the energy expectation value to oscillate. This is the result of a rigorous solution to the time-dependent Schrödinger equation describing the coupling dynamics of two spins 1/2 in an external field. Demanding to solve a four-dimensional system of coupled linear differential equations of first order, the bulkiness of even this simplest case already brings the calculations to the verge of becoming infeasible analytically.