In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly. The name derives from the German mathematicians Alfred Clebsch (1833–1872) and Paul Gordan (1837–1912), who encountered an equivalent problem in invariant theory.
In terms of classical mathematics, the CG coefficients, or at least those associated to the group SO(3), may be defined much more directly, by means of formulae for the multiplication of spherical harmonics. The addition of spins in quantum-mechanical terms can be read directly from this approach. The formulas below use Dirac's bra-ket notation.
Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis.
Below, this definition is made precise by defining angular momentum operators, angular momentum eigenstates, and tensor products of these states.
From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found. To find numerical values for the coefficients a phase convention must be adopted. Below the Condon–Shortley phase convention is chosen.
- 1 Angular momentum operators
- 2 Angular momentum states
- 3 Tensor product space
- 4 Formal definition of Clebsch–Gordan coefficients
- 5 Recursion relations
- 6 Explicit expression
- 7 Orthogonality relations
- 8 Special cases
- 9 Symmetry properties
- 10 Relation to 3-jm symbols
- 11 Relation to Wigner D-matrices
- 12 Other Properties
- 13 SU(N) Clebsch–Gordan coefficients
- 14 See also
- 15 References
- 16 External links
- 17 Further reading
Angular momentum operators
where is the Levi-Civita symbol. Together the three operators define a "vector operator":
By developing this concept further, one can define an operator as an "inner product" of with itself:
It is an example of a Casimir operator.
We also define raising () and lowering () operators:
Angular momentum states
It can be shown from the above definitions that commutes with , and
When two Hermitian operators commute, a common set of eigenfunctions exists. Conventionally and are chosen. From the commutation relations the possible eigenvalues can be found. The result is:
The raising and lowering operators change the value of :
A (complex) phase factor could be included in the definition of . The choice made here is in agreement with the Condon and Shortley phase conventions. The angular momentum states must be orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and are assumed to be normalized:
Note that the italicized and denote integers or half-integers, which label the total angular momentum of the system (e.g. for an electron and for a photon). On the other hand, the roman and denote operators (the last being a vector operator).
Tensor product space
Let be the -dimensional vector space spanned by the states
and the -dimensional vector space spanned by
The tensor product of these spaces, , has a -dimensional uncoupled basis
Angular momentum operators acting on can be defined by
Total angular momentum operators are defined by
The total angular momentum operators satisfy the required commutation relations
and hence total angular momentum eigenstates exist:
It can be derived that the total angular momentum quantum number must satisfy the triangular condition
The total number of total angular momentum eigenstates is equal to the dimension of :
The total angular momentum states form an orthonormal basis of :
Formal definition of Clebsch–Gordan coefficients
The total angular momentum states can be expanded with the use of the completeness relation in the uncoupled basis
The expansion coefficients are called Clebsch–Gordan coefficients.
Applying the operator
to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when
The recursion relations were discovered by physicist Giulio Racah from the Hebrew University of Jerusalem. Applying the total angular momentum raising and lowering operators
to the left hand side of the defining equation gives
Applying the same operators to the right hand side gives
Combining these results gives recursion relations for the Clebsch–Gordan coefficients
Taking the upper sign with gives
In the Condon and Shortley phase convention the coefficient is taken real and positive. With the last equation all other Clebsch–Gordan coefficients can be found. The normalization is fixed by the requirement that the sum of the squares, which corresponds to the norm of the state must be one.
The lower sign in the recursion relation can be used to find all the Clebsch–Gordan coefficients with . Repeated use of that equation gives all coefficients.
This procedure to find the Clebsch–Gordan coefficients shows that they are all real (in the Condon and Shortley phase convention).
For an explicit expression of the Clebsch–Gordan coefficients and tables with numerical values, see table of Clebsch–Gordan coefficients.
These are most clearly written down by introducing the alternative notation
The first orthogonality relation is
(using the completeness relation that )and the second
For the Clebsch–Gordan coefficients are given by
For and we have
For and we have
For we have
For we have
A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to 3-jm symbols using the equation given below. The symmetry properties of 3-jm symbols are much simpler. Care is needed when simplifying phase factors, because the quantum numbers can be integer or half integer, e.g., is equal to 1 for integer and equal to −1 for half-integer . The following relations, however, are valid in either case:
and for , and appearing in the same Clebsch–Gordan coefficient:
Relation to 3-jm symbols
Clebsch–Gordan coefficients are related to 3-jm symbols which have more convenient symmetry relations.
Relation to Wigner D-matrices
SU(N) Clebsch–Gordan coefficients
For arbitrary groups and their representations, Clebsch–Gordan coefficients are not known in general. However, algorithms to produce Clebsch–Gordan coefficients for the Special unitary group are known. A web interface for tabulating SU(N) Clebsch–Gordan coefficients is readily available.
- 3-jm symbol
- Racah W-coefficient
- 6-j symbol
- 9-j symbol
- Spherical harmonics
- Spherical basis
- Associated Legendre polynomials
- Angular momentum
- Angular momentum coupling
- Total angular momentum quantum number
- Azimuthal quantum number
- Table of Clebsch–Gordan coefficients
- Wigner D-matrix
- Angular momentum diagrams (quantum mechanics)
- Alex, A.; M. Kalus, A. Huckleberry, and J. von Delft (February 2011). "A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients". J. Math. Phys. 82: 023507. arXiv:1009.0437. Bibcode:2011JMP....52b3507A. doi:10.1063/1.3521562. Retrieved 2011-04-13.
- PDF Table of Clebsch–Gordan Coefficients, Spherical Harmonics, and d-Functions
- Clebsch–Gordan, 3-j and 6-j Coefficient Web Calculator
- Downloadable Clebsch–Gordan Coefficient Calculator for Mac and Windows
- Web interface for tabulating SU(N) Clebsch–Gordan coefficients
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