Chapter 8: CLEBSCH-GORDAN COEFFICIENTS AND 3jm SYMBOLS

The following sections are included:

• DEFINITION

  • The Clebsch-Gordan Coefficients

  • The Wigner 3jm Symbols

  • Regge R-Symbols

• EXPLICIT FORMS OF THE CLEBSCH-GORDAN COEFFICIENTS AND THEIR RELATIONS TO OTHER FUNCTIONS

  • Representations of the Clebsch-Gordan Coefficients in the Form of Algebraic Sums

  • Quasi-Binomial Representation of the Clebsch-Gordan Coefficients

  • Clebsch-Gordan Coefficients and Finite Differences

  • Expressions for the Clebsch-Gordan Coefficients in Terms of the Binomial Coefficients [126]

  • Representations of the Clebsch-Gordan Coefficients in Terms of of the Hypergeometric Functions

  • Representations of the 3jm Symbols in the Form of Algebraic Sums

  • Quasi-binomial Representations of the 3jm Symbols

• INTEGRAL REPRESENTATIONS

  • Integrals Involving Algebraic Functions

  • Integrals Involving the Wigner D-Punctions

  • Integral Involving the Spherical Harmonics

  • Integral Representations for Products of the Clebsch-Gordan Coefficients

• SYMMETRY PROPERTIES

  • Symmetry Properties of the R-Symbols

  • Symmetry Properties of the 3jm Symbols

  • Symmetry Properties of the Clebsch-Gordan Coefficients

  • “Mirror” Symmetry

  • Properties of the Vector-Addition Coefficients under Transformations of the Coordinate System and Time Reversal

• EXPLICIT FORMS OF THE CLEBSCH-GORDAN COEFFICIENTS FOR SPECIAL VALUES OF THE ARGUMENTS

  • Special Values of Momenta a, b, c

  • Special Values of Momentum Projections

• RECURSION RELATIONS FOR THE CLEBSCH-GORDAN COEFFICIENTS

  • General Recursion Relations

  • Arguments α,β, γ Change by 1

  • Arguments Change by 1/2

  • The Case When α = β = γ = 0

  • Arguments a, b, c Change by 1

  • Arguments a, b, α, β Change by 1

  • Arguments c, b, γ, β Change by 1

  • Recursion Relations for the Regge Symbols

• SUMS OF PRODUCTS OF THE CLEBSCH-GORDAN COEFFICIENTS

  • Sums Involving One Clebsch-Gordan Coefficient

  • Sums Involving Products of Two Clebsch-Gordan Coefficients

  • Sums Involving Products of Three Clebsch-Gordan Coefficients

  • Sums Involving Products of Four Clebsch-Gordan Coefficients

  • Sums Involving Products of the Clebsch-Gordan Coefficients and One 6j Symbol

  • Sums Involving Products of the Clebsch-Gordan Coefficients and One 9j Symbol

  • Some Additional Sums of Products of Two Clebsch-Gordan Coefficients

• GENERATING FUNCTIONS

  • Regge Determinant to Power J

  • Products of Binomials

  • Exponential Function

  • Hypergeometric Function

  • Wigner D-Function

  • Generating Function for the Coefficients $C_{a0b0}^{c0}$

• CLASSICAL LIMIT AND ASYMPTOTIC EXPRESSIONS FOR THE CLEBSCH-GORDAN COEFFICIENTS

  • Asymptotic Expressions for a, c ≫ b

  • Asymptotic Expressions for a, b, c, γ ≫ a + b − γ

  • Semiclassical Formulas for a, b, c ≫ 1

  • Squares of the Clebsch-Gordan Coefficients in the Classical Limit

• ZEROS OF THE VECTOR-ADDITION COEFFICIENTS

• CONNECTION OF THE CLEBSCH-GORDAN COEFFICIENTS AND THE 3jm SYMBOLS WITH ANALOGOUS FUNCTIONS OF OTHER AUTHORS

• ALGEBRAIC TABLES OF THE CLEBSCH-GORDAN COEFFICIENTS

• NUMERICAL TABLES OF THE CLEBSCH-GORDAN COEFFICIENTS