9
Angular Mome
2. Angular Momentum Theory
Angular momentum theory is presented from the
viewpoint of the group SU(1) of unimodular unitary
matrices of order two. This is the basic quantum
mechanical rotation group for implementing the
consequences of rotational symmetry into isolated
complex physical systems, and gives the structure
of the angular momentum multiplets of such
systems. This entails the study of representation
functions of SU(2), the Lie algebra of SU(2) and
copies thereof, and the associated Wigner–
Clebsch–Gordan coefficients, Racah coefficients,
and 1n–j coefficients, with an almost boundless
set of inter-relations, and presentations of the
associated conceptual framework. The relationship
to the rotation group in physical 3-space is
given in detail. Formulas are often given in
a compendium format with brief introductions
on their physical and mathematical content.
A special effort is made to inter-relate the
material to the special functions of mathematics
and to the combinatorial foundations of the
subject.
2.1 Orbital Angular Momentum .................. 12
2.1.1 Cartesian Representation ........... 12
2.1.2 Spherical Polar Coordinate
Representation ......................... 15
2.2 Abstract Angular Momentum ................ 16
2.3 Representation Functions..................... 18
2.3.1 Parametrizations
of the Groups SU(2) and SO(3,R)... 18
2.3.2 Explicit Forms
of Representation Functions ....... 19
2.3.3 Relations to Special Functions..... 21
2.3.4 Orthogonality Properties ............ 21
2.3.5 Recurrence Relations ................. 22
2.3.6 Symmetry Relations................... 23
2.4 Group and Lie Algebra Actions .............. 25
2.4.1 Matrix Group Actions ................. 25
2.4.2 Lie Algebra Actions .................... 26
2.4.3 Hilbert Spaces ........................... 26
2.4.4 Relation
to Angular Momentum Theory..... 26
2.5 Differential Operator Realizations
of Angular Momentum ......................... 28
2.6 The Symmetric Rotor
and Representation Functions .............. 29
2.7 Wigner–Clebsch–Gordan
and 3-j Coefficients ............................. 31
2.7.1 Kronecker Product Reduction...... 32
2.7.2 Tensor Product Space
Construction ............................. 33
2.7.3 Explicit Forms
of WCG-Coefficients ................... 33
2.7.4 Symmetries of WCG-Coefficients
in 3-j Symbol Form ................... 35
2.7.5 Recurrence Relations ................. 36
2.7.6 Limiting Properties
and Asymptotic Forms ............... 36
2.7.7 WCG-Coefficients as Discretized
Representation Functions........... 37
2.8 Tensor Operator Algebra....................... 37
2.8.1 Conceptual Framework .............. 37
2.8.2 Universal Enveloping Algebra
of J ......................................... 38
2.8.3 Algebra
of Irreducible Tensor Operators ... 39
2.8.4 Wigner–Eckart Theorem ............. 39
2.8.5 Unit Tensor Operators
or Wigner Operators................... 40
2.9 Racah Coefficients ............................... 43
2.9.1 Basic Relations Between WCG
and Racah Coefficients............... 43
2.9.2 Orthogonality and Explicit Form .. 43
2.9.3 The Fundamental Identities
Between Racah Coefficients ........ 44
2.9.4 Schwinger–Bargmann
Generating Function
and its Combinatorics ................ 44
2.9.5 Symmetries of 6–j Coefficients.... 45
2.9.6 Further Properties ..................... 46
2.10 The 9–j Coefficients ............................. 47
2.10.1 Hilbert Space
and Tensor Operator Actions ....... 47
2.10.2 9–j Invariant Operators ............. 47
2.10.3 Basic Relations Between 9–j
Coefficients and 6–j Coefficients . 48
Part A 2