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:2411.03952 · Journal of Physics Conference Series

Generalized exchange operators for a system of spin-1 particles

Charlie Jeudy, Michel Rouleux

Published Dec 1, 2024• 0 citations
arXivHTMLPDF

Abstract

Abstract The irreps ( S U ( 2 ) , H , U ) of SU(2) of dimension (2S + 1)N, i.e. operators acting on the space H = H N = C ( 2 S + 1 ) N of N identical particles with spin S, are described by Clebsch-Gordan decomposition into inequivalent irreps. In the special case S = 1/2, Dirac [1] discovered that there is another rep given by ( S ( N ) , H , V ) where S ( N ) is the permutation group, Thus, the standard “linear” Hamiltonian, or Heisenberg interaction Hamiltonian H 0 = ∑ 1 ≤ i ≤ N S → i ⋅ S → j , where σ → i = 2 S → i is the vector of Pauli matrices, can be interpreted as the sum of the “Exchange Operators” Pij between particles i and j. Schrödinger [3] generalized to higher spin numbers S the Exchange Operator

DOI
10.1088/1742-6596/2912/1/012043

Related papers

Generalized Dirac identities and explicit relations between the permutational symmetry and the spin operators for systems of identical particles

Jacob Katriel, Josef Paldus, Reuben Pauncz

Abstract The well known one‐to‐one correspondence between the eigenstates of the total spin for a system of spin‐½ particles and irreducible representations of the symmetric group with up to two rows in the Young shape is the basis of interesting formal developments in quantum chemistry and in the theory of magnetism. As an explicit manifestation of this correspondence the class operators of the symmetric group are demonstrated to be expressible in terms of the total spin operator. This correspondence does not hold for higher elementary spins. The extension to arbitrary spin is investigated using Schrödinger's generalization of the Dirac identity, which expresses the transpositions in terms of two‐particle spin operators. It is shown that additional operators, which for σ = ½ reduce to the total spin operator, are needed for a complete classification. Some aspects of the formalism are developed in detail for σ = 1. In this case a classification identical with that provided by the irreducible representations of the symmetric group is obtained in terms of the eigenstates of two commuting operators, one of which is the total spin operator.

International Journal of Quantum Chemistry•Aug 1, 1985•14 citations

Permutational symmetry of many-particle spin-functions: The dirac identities

Jacob Katriel

Explicit expressions are derived for the action of all conjugacy classsums of the symmetric group within the space spanned by n identical spin-1/2 particles, in terms of the total spin operator. For identical particles with elementary spin σ all conjugacy class-sums are presented in terms of the lowest 2σ single cycle conjugacy class-sums.

AIP conference proceedings•Jan 1, 2000•0 citations

Group projector generalization of the Dirac-Heisenberg model

Milan Damnjanović

The general form of the operators commuting with the ground representation (appearing in many physical problems within the single-particle approximation) of the group is found. With the help of the modified group projector technique, this result is applied to a system of identical particles with spin-independent interaction, to derive the Dirac-Heisenberg Hamiltonian and its effective space for arbitrary orbital occupation numbers and arbitrary spin. This gives transparent insight into the physical contents of this Hamiltonian, showing that formal generalizations with spin greater than ½ involve non-trivial additional physical assumptions.

arXiv:cond-mat/0004366•Journal of Physics A Mathematical and General•Mar 10, 2000•1 citation

A Spinor Approach to the SU(2) Clebsch-Gordan Coefficients*

Ling-Di Yang, Famin Chen

Abstract We review the irreducible representation of an angular momentum vector operator constructed in terms of spinor algebra. We generalize the idea of spinor approach to study the coupling of the eigenstates of two independent angular momentum vector operators. Utilizing the spinor algebra, we are able to develop a simple way for calculating the SU(2) Clebsch-Gordan (CG) coefficients. The explicit expression for the SU(2) CG coefficients is worked out, and some simple physical examples are presented to illustrate the spinor approach.

Communications in Theoretical Physics•Jun 1, 2019•0 citations

The universal Hamiltonian of the exchange interaction for the system of particles with an arbitrary spin j

E. V. Orlenko

Abstract On the basis of the arrangement symmetry for the system of identical particles the universal Hamiltonian of the exchange interaction in the spin representation is developed. For the case of spin‐1/2 particles the obtained Hamiltonian is equivalent to the Heisenberg form. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

International Journal of Quantum Chemistry•Jan 1, 2007•7 citations

On a Complete Basis in the Space of Rotationally Invariant Operators of $$N$$ Quantum Spins $$1/2$$

Filipp Gennadevich Uskov

Systems of quantum spins $$1/2$$ with isotropic Heisenberg interaction play an important role in physics. In studying such systems, it may be useful to have a complete, yet non-overcomplete, basis of operators each of which has the symmetry of the Hamiltonian, i.e., is invariant with respect to rotations (global $$\mathrm{SU}(2)$$ transformations of the Pauli matrices). This paper presents an algorithm for constructing such a basis. The algorithm is implemented in Wolfram Mathematica.

Proceedings of the Steklov Institute of Mathematics•Jul 1, 2021•1 citation
Abstract The irreps ( S U ( 2 ) , H , U ) of SU(2) of dimension (2S + 1)N, i.e. operators acting on the space H = H N = C ( 2 S + 1 ) N of N identical particles with spin S, are described by Clebsch-Gordan decomposition into inequivalent irreps. In the special case S = 1/2, Dirac [1] discovered that there is another rep given by ( S ( N ) , H , V ) where S ( N ) is the permutation group, Thus, the standard “linear” Hamiltonian, or Heisenberg interaction Hamiltonian H 0 = ∑ 1 ≤ i ≤ N S → i ⋅ S → j , where σ → i = 2 S → i is the vector of Pauli matrices, can be interpreted as the sum of the “Exchange Operators” Pij between particles i and j. Schrödinger [3] generalized to higher spin numbers S the Exchange Operator
Abstract The well known one‐to‐one correspondence between the eigenstates of the total spin for a system of spin‐½ particles and irreducible representations of the symmetric group with up to two rows in the Young shape is the basis of interesting formal developments in quantum chemistry and in the theory of magnetism. As an explicit manifestation of this correspondence the class operators of the symmetric group are demonstrated to be expressible in terms of the total spin operator. This correspondence does not hold for higher elementary spins. The extension to arbitrary spin is investigated using Schrödinger's generalization of the Dirac identity, which expresses the transpositions in terms of two‐particle spin operators. It is shown that additional operators, which for σ = ½ reduce to the total spin operator, are needed for a complete classification. Some aspects of the formalism are developed in detail for σ = 1. In this case a classification identical with that provided by the irreducible representations of the symmetric group is obtained in terms of the eigenstates of two commuting operators, one of which is the total spin operator.
The general form of the operators commuting with the ground representation (appearing in many physical problems within the single-particle approximation) of the group is found. With the help of the modified group projector technique, this result is applied to a system of identical particles with spin-independent interaction, to derive the Dirac-Heisenberg Hamiltonian and its effective space for arbitrary orbital occupation numbers and arbitrary spin. This gives transparent insight into the physical contents of this Hamiltonian, showing that formal generalizations with spin greater than ½ involve non-trivial additional physical assumptions.
Abstract We review the irreducible representation of an angular momentum vector operator constructed in terms of spinor algebra. We generalize the idea of spinor approach to study the coupling of the eigenstates of two independent angular momentum vector operators. Utilizing the spinor algebra, we are able to develop a simple way for calculating the SU(2) Clebsch-Gordan (CG) coefficients. The explicit expression for the SU(2) CG coefficients is worked out, and some simple physical examples are presented to illustrate the spinor approach.
On a Complete Basis in the Space of Rotationally Invariant Operators of NNN Quantum Spins 1/21/21/2
Systems of quantum spins 1/21/21/2 with isotropic Heisenberg interaction play an important role in physics. In studying such systems, it may be useful to have a complete, yet non-overcomplete, basis of operators each of which has the symmetry of the Hamiltonian, i.e., is invariant with respect to rotations (global SU(2)\mathrm{SU}(2)SU(2) transformations of the Pauli matrices). This paper presents an algorithm for constructing such a basis. The algorithm is implemented in Wolfram Mathematica.