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 Physics A Mathematical and General

Addition of a large number of identical angular momenta. Statistical distributions

V V Mikhailov

Published Dec 1, 1979• 3 citations
Publisher

Abstract

The quantum mechanical addition of an arbitrary (very large) number of identical angular momenta is considered. The resulting states jm usually occur many times. The corresponding multiplicities P j and Q m are investigated. Several explicit formulae for Q m are given. The asymptotic approximations for multiplicities P j and Q m , which are the Wigner and Gauss distributions respectively, are derived. All parameters of the distributions are explicitly obtained. A comparison of theoretical multiplicities with the exact ones is made when the number of angular moments is relatively small.

DOI
10.1088/0305-4470/12/12/011

Related papers

Addition or arbitrary number of identical angular momenta

V V Mihailov

Under the vector addition of angular momenta, each equal to s, the resulting angular momentaj occur with some multiplicity. The general formula for these multiplicities is obtained by a simple algebraic method. In the case of s=1/2 the expression proves to be a well known one. Also given are some relations connected with the multiplicities.

Journal of Physics A Mathematical and General•Feb 1, 1977•29 citations

Addition of arbitrary number of identical angular momenta

Mohammad A. Rashid

An alternate method is used to establish the analytic expression recently given and incompletely proved by Mikhailov (see ibid., vol.10, p.147 (1977)) for the multiplicity of occurrence of any angular momentum in the vector addition of n identical angular momenta.

Journal of Physics A Mathematical and General•Aug 1, 1977•9 citations

Semiclassical approximations to 3j- and 6j-coefficients for quantum-mechanical coupling of angular momenta

Klaus J. Schulten, Roy Gerald Gordon

The coupling of angular momenta is studied using quantum mechanics in the limit of large quantum numbers (semiclassical limit). Uniformly valid semiclassical expressions are derived for the 3j (Wigner) coefficients coupling two angular momenta, and for the 6j (Racah) coefficients coupling three angular momenta. In three limiting cases our new expressions reduce to those conjectured by Ponzano and Regge. The derivation involves solving the recursion relations satisfied by these coefficients, by a discrete analog of the WKB method. Terms of the order of the inverse square of the quantum numbers are neglected in the derivation, so that the results should be increasingly accurate for larger angular momenta. Numerical results confirm this asymptotic convergence. Moreover, the results are of a useful accuracy even at small quantum numbers.

Journal of Mathematical Physics•Oct 1, 1975•136 citations

Complementarity in Angular-Momentum States

Bernard A. Lippmann

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.

Physical Review•Sep 1, 1953•0 citations

Matrix Representation of the Angular Momentum Projection Operator

Joseph Y. Shapiro

Formulas are obtained which give the matrix representation, relative to a product basis, of the projection operator for the total angular momentum of a system. If the individual angular momenta are not too large, the matrix elements depend upon a small number of parameters, independent of the number of angular momenta coupled. Recurrence relations between elements and symmetry properties of elements are derived. These results enable one to perform the vector coupling of a large number of angular momenta in a relatively simple fashion. The connection between the matrix elements and vector coupling coefficients is discussed. Several important special cases are treated in detail.

Journal of Mathematical Physics•Nov 1, 1965•44 citations

Relationship of the commutation rules to classical-like representations of quantum angular momenta addition

A. Morales, Alejandro Amaya-Tapia

We perform an analysis of a graphical representation for the addition of two angular momenta, focusing our attention on the angle δ between the xy components of two angular momenta. Then we propose a new complete set of commuting observables, which differ from the usual sets that are connected by the Clebsch–Gordan coefficients. This set shows that the angle δ can be a well-defined variable in quantum mechanics. An empirical analysis of the graphical representations of the angular momenta relations, which may include the angle δ, followed by quantum mechanical considerations, leads to the vanishing of certain quantum angular momentum commutators for specific states. Therefore, although the commutators are not null in general, the quantum addition of angular momenta may be represented using classical-like diagrams.

American Journal of Physics•Nov 1, 1999•0 citations
The quantum mechanical addition of an arbitrary (very large) number of identical angular momenta is considered. The resulting states jm usually occur many times. The corresponding multiplicities P j and Q m are investigated. Several explicit formulae for Q m are given. The asymptotic approximations for multiplicities P j and Q m , which are the Wigner and Gauss distributions respectively, are derived. All parameters of the distributions are explicitly obtained. A comparison of theoretical multiplicities with the exact ones is made when the number of angular moments is relatively small.
The coupling of angular momenta is studied using quantum mechanics in the limit of large quantum numbers (semiclassical limit). Uniformly valid semiclassical expressions are derived for the 3j (Wigner) coefficients coupling two angular momenta, and for the 6j (Racah) coefficients coupling three angular momenta. In three limiting cases our new expressions reduce to those conjectured by Ponzano and Regge. The derivation involves solving the recursion relations satisfied by these coefficients, by a discrete analog of the WKB method. Terms of the order of the inverse square of the quantum numbers are neglected in the derivation, so that the results should be increasingly accurate for larger angular momenta. Numerical results confirm this asymptotic convergence. Moreover, the results are of a useful accuracy even at small quantum numbers.
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.
Formulas are obtained which give the matrix representation, relative to a product basis, of the projection operator for the total angular momentum of a system. If the individual angular momenta are not too large, the matrix elements depend upon a small number of parameters, independent of the number of angular momenta coupled. Recurrence relations between elements and symmetry properties of elements are derived. These results enable one to perform the vector coupling of a large number of angular momenta in a relatively simple fashion. The connection between the matrix elements and vector coupling coefficients is discussed. Several important special cases are treated in detail.
We perform an analysis of a graphical representation for the addition of two angular momenta, focusing our attention on the angle δ between the xy components of two angular momenta. Then we propose a new complete set of commuting observables, which differ from the usual sets that are connected by the Clebsch–Gordan coefficients. This set shows that the angle δ can be a well-defined variable in quantum mechanics. An empirical analysis of the graphical representations of the angular momenta relations, which may include the angle δ, followed by quantum mechanical considerations, leads to the vanishing of certain quantum angular momentum commutators for specific states. Therefore, although the commutators are not null in general, the quantum addition of angular momenta may be represented using classical-like diagrams.