Clebsch–Gordan coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. 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 (i.e., a reducible representation) 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 and Paul Gordan, who encountered an equivalent problem in invariant theory.
From a vector calculus perspective, the CG coefficients associated with the SO(3) group can be defined simply in terms of integrals of products of spherical harmonics and their complex conjugates. The addition of spins in quantummechanical terms can be read directly from this approach as spherical harmonics are eigenfunctions of total angular momentum and projection thereof onto an axis, and the integrals correspond to the Hilbert space inner product.^{[1]} From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found. There also exist complicated explicit formulas for their direct calculation.^{[2]}
The formulas below use Dirac's bra–ket notation and the Condon–Shortley phase convention^{[3]} is adopted.
Contents
 1 Angular momentum operators
 2 Spherical basis for angular momentum eigenstates
 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 Wigner 3j symbols
 11 Relation to Wigner Dmatrices
 12 Relation to spherical harmonics
 13 Other Properties
 14 SU(n) Clebsch–Gordan coefficients
 15 See also
 16 Remarks
 17 Notes
 18 References
 19 External links
 20 Further reading
Angular momentum operatorsEdit
Angular momentum operators are selfadjoint operators j_{x}, j_{y}, and j_{z} that satisfy the commutation relations
where ε_{klm} is the LeviCivita symbol. Together the three operators define a vector operator, a rank one Cartesian tensor operator,
It also known as a spherical vector, since it is also a spherical tensor operator. It is only for rank one that spherical tensor operators coincide with the Cartesian tensor operators.
By developing this concept further, one can define another operator j^{2} as the inner product of j with itself:
This is an example of a Casimir operator. It is diagonal and its eigenvalue characterizes the particular irreducible representation of the angular momentum algebra so(3) ≅ su(2). This is physically interpreted as the square of the total angular momentum of the states on which the representation acts.
One can also define raising (j_{+}) and lowering (j_{−}) operators, the socalled ladder operators,
Spherical basis for angular momentum eigenstatesEdit
It can be shown from the above definitions that j^{2} commutes with j_{x}, j_{y}, and j_{z}:
When two Hermitian operators commute, a common set of eigenstates exists. Conventionally, j^{2} and j_{z} are chosen. From the commutation relations, the possible eigenvalues can be found. These eigenstates are denoted j m⟩ where j is the angular momentum quantum number and m is the angular momentum projection onto the zaxis.
They comprise the spherical basis, are complete, and satisfy the following eigenvalue equations,
The raising and lowering operators can be used to alter the value of m,
where the ladder coefficient is given by:

(1)
In principle, one may also introduce a (possibly complex) phase factor in the definition of . The choice made in this article is in agreement with the Condon–Shortley phase convention. The angular momentum states are orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and are assumed to be normalized,
Here the italicized j and m denote integer or halfinteger angular momentum quantum numbers of a particle or of a system. On the other hand, the roman j_{x}, j_{y}, j_{z}, j_{+}, j_{−}, and j^{2} denote operators. The symbols are Kronecker deltas.
Tensor product spaceEdit
We now consider systems with two physically different angular momenta j_{1} and j_{2}. Examples include the spin and the orbital angular momentum of a single electron, or the spins of two electrons, or the orbital angular momenta of two electrons. Mathematically, this means that the angular momentum operators act on a space of dimension and also on a space of dimension . We are then going to define a family of "total angular momentum" operators acting on the tensor product space , which has dimension . The action of the total angular momentum operator on this space constitutes a representation of the su(2) Lie algebra, but a reducible one. The reduction of this reducible representation into irreducible pieces is the goal of Clebsch–Gordan theory.
Let V_{1} be the (2 j_{1} + 1)dimensional vector space spanned by the states
 ,
and V_{2} the (2 j_{2} + 1)dimensional vector space spanned by the states
 .
The tensor product of these spaces, V_{3} ≡ V_{1} ⊗ V_{2}, has a (2 j_{1} + 1) (2 j_{2} + 1)dimensional uncoupled basis
 .
Angular momentum operators are defined to act on states in V_{3} in the following manner:
and
where 1 denotes the identity operator.
The total^{[nb 1]} angular momentum operators are defined by the coproduct (or tensor product) of the two representations acting on V_{1}⊗V_{2},
The total angular momentum operators can be shown to satisfy the very same commutation relations,
where k, l, m ∈ {x, y, z}. Indeed, the preceding construction is the standard method^{[4]} for constructing an action of a Lie algebra on a tensor product representation.
Hence, a set of coupled eigenstates exist for the total angular momentum operator as well,
for M ∈ {−J, −J + 1, …, J}. Note that it is common to omit the [j_{1} j_{2}] part.
The total angular momentum quantum number J must satisfy the triangular condition that
 ,
such that the three nonnegative integer or halfinteger values could correspond to the three sides of a triangle.^{[5]}
The total number of total angular momentum eigenstates is necessarily equal to the dimension of V_{3}:
As this computation suggests, the tensor product representation decomposes as the direct sum of one copy of each of the irreducible representations of dimension , where ranges from to in increments of 1.^{[6]} As an example, consider the tensor product of the threedimensional representation corresponding to with the twodimensional representation with . The possible values of are then and . Thus, the sixdimensional tensor product representation decomposes as the direct sum of a twodimensional representation and a fourdimensional representation.
The goal is now to describe the preceding decomposition explicitly, that is, to explicitly describe basis elements in the tensor product space for each of the component representations that arise.
The total angular momentum states form an orthonormal basis of V_{3}:
These rules may be iterated to, e.g., combine n doublets (s=1/2) to obtain the ClebschGordan decomposition series, (Catalan's triangle),
where is the integer floor function; and the number preceding the boldface irreducible representation dimensionality (2j+1) label indicates multiplicity of that representation in the representation reduction.^{[7]} For instance, from this formula, addition of three spin 1/2s yields a spin 3/2 and two spin 1/2s, .
Formal definition of Clebsch–Gordan coefficientsEdit
The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis

(2)
The expansion coefficients
are the Clebsch–Gordan coefficients. Note that some authors write them in a different order such as ⟨j_{1} j_{2}; m_{1} m_{2}J M⟩. Another common notation is
⟨j_{1} m_{1} j_{2} m_{2}  J M⟩ = C^{JM}
_{j1m1j2m2}.
Applying the operator
to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when
 .
Recursion relationsEdit
The recursion relations were discovered by physicist Giulio Racah from the Hebrew University of Jerusalem in 1941.
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
where C_{±} was defined in 1. Combining these results gives recursion relations for the Clebsch–Gordan coefficients:
 .
Taking the upper sign with the condition that M = J gives initial recursion relation:
 .
In the Condon–Shortley phase convention, one adds the constraint that
(and is therefore also real).
The Clebsch–Gordan coefficients ⟨j_{1} m_{1} j_{2} m_{2}  J M⟩ can then be found from these recursion relations. The normalization is fixed by the requirement that the sum of the squares, which equivalent to the requirement that the norm of the state [j_{1} j_{2}] J J⟩ must be one.
The lower sign in the recursion relation can be used to find all the Clebsch–Gordan coefficients with M = J − 1. 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–Shortley phase convention.
Explicit expressionEdit
Orthogonality relationsEdit
These are most clearly written down by introducing the alternative notation
The first orthogonality relation is
(derived from the fact that 1 ≡ ∑_{x} x⟩ ⟨x) and the second one is
 .
Special casesEdit
For J = 0 the Clebsch–Gordan coefficients are given by
 .
For J = j_{1} + j_{2} and M = J we have
 .
For j_{1} = j_{2} = J / 2 and m_{1} = −m_{2} we have
 .
For j_{1} = j_{2} = m_{1} = −m_{2} we have
For j_{2} = 1, m_{2} = 0 we have
For j_{2} = 1/2 we have
Symmetry propertiesEdit
A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to Wigner 3j symbols using 3. The symmetry properties of Wigner 3j symbols are much simpler.
Rules for phase factorsEdit
Care is needed when simplifying phase factors: a quantum number may be a halfinteger rather than an integer, therefore (−1)^{2k} is not necessarily 1 for a given quantum number k unless it can be proven to be an integer. Instead, it is replaced by the following weaker rule:
for any angularmomentumlike quantum number k.
Nonetheless, a combination of j_{i} and m_{i} is always an integer, so the stronger rule applies for these combinations:
This identity also holds if the sign of either j_{i} or m_{i} or both is reversed.
It is useful to observe that any phase factor for a given (j_{i}, m_{i}) pair can be reduced to the canonical form:
where a ∈ {0, 1, 2, 3} and b ∈ {0, 1} (other conventions are possible too). Converting phase factors into this form makes it easy to tell whether two phase factors are equivalent. (Note that this form is only locally canonical: it fails to take into account the rules that govern combinations of (j_{i}, m_{i}) pairs such as the one described in the next paragraph.)
An additional rule holds for combinations of j_{1}, j_{2}, and j_{3} that are related by a ClebschGordan coefficient or Wigner 3j symbol:
This identity also holds if the sign of any j_{i} is reversed, or if any of them are substituted with an m_{i} instead.
Relation to Wigner 3j symbolsEdit
Clebsch–Gordan coefficients are related to Wigner 3j symbols which have more convenient symmetry relations.

(3)
The factor (−1)^{2 j2} is due to the Condon–Shortley constraint that ⟨j_{1} j_{1} j_{2} (J − j_{1})J J⟩ > 0, while (–1)^{J − M} is due to the timereversed nature of J M⟩.
Relation to Wigner DmatricesEdit
Relation to spherical harmonicsEdit
In the case where integers are involved, the coefficients can be related to integrals of spherical harmonics:
It follows from this and orthonormality of the spherical harmonics that CG coefficients are in fact the expansion coefficients of a product of two spherical harmonics in terms a single spherical harmonic:
Other PropertiesEdit
SU(n) Clebsch–Gordan coefficientsEdit
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.^{[8]}^{[9]} In particular, SU(3) ClebschGordan coefficients have been computed and tabulated because of their utility in characterizing hadronic decays, where a flavorSU(3) symmetry exists that relates the up, down, and strange quarks.^{[10]}^{[11]} A web interface for tabulating SU(N) Clebsch–Gordan coefficients is readily available.
See alsoEdit
 3j symbol
 6j symbol
 9j symbol
 Racah Wcoefficient
 Spherical harmonics
 Spherical basis
 Tensor products of representations
 Associated Legendre polynomials
 Angular momentum
 Angular momentum coupling
 Total angular momentum quantum number
 Azimuthal quantum number
 Table of Clebsch–Gordan coefficients
 Wigner Dmatrix
 Wigner–Eckart theorem
 Angular momentum diagrams (quantum mechanics)
 Clebsch–Gordan coefficient for SU(3)
 LittlewoodRichardson coefficient
RemarksEdit
 ^ The word "total" is often overloaded to mean several different things. In this article, "total angular momentum" refers to a generic sum of two angular momentum operators j_{1} and j_{2}. It is not to be confused with the other common use of the term "total angular momentum" that refers specifically to the sum of orbital angular momentum and spin.
NotesEdit
 ^ Greiner & Müller 1994
 ^ Edmonds 1957
 ^ Condon & Shortley 1970
 ^ Hall 2015 Section 4.3.2
 ^ Merzbacher 1998
 ^ Hall 2015 Appendix C
 ^ Zachos, C K (1992). "Altering the Symmetry of Wavefunctions in Quantum Algebras and Supersymmetry". Modern Physics Letters. A7 (18): 1595–1600. arXiv:hepth/9203027. Bibcode:1992MPLA....7.1595Z. doi:10.1142/S0217732392001270.
 ^ Alex et al. 2011
 ^ Kaplan & Resnikoff 1967
 ^ de Swart 1963
 ^ Kaeding 1995
ReferencesEdit
 Alex, A.; Kalus, M.; Huckleberry, A.; von Delft, J. (2011). "A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients". J. Math. Phys. 82 (2): 023507. arXiv:1009.0437. Bibcode:2011JMP....52b3507A. doi:10.1063/1.3521562.
 Condon, Edward U.; Shortley, G. H. (1970). "Ch. 3". The Theory of Atomic Spectra. Cambridge: Cambridge University Press. ISBN 9780521092098.
 Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton, New Jersey: Princeton University Press. ISBN 9780691079127.
 Greiner, Walter; Müller, Berndt (1994). Quantum Mechanics: Symmetries (2nd ed.). Springer Verlag. ISBN 9783540580805.
 Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 9783319134666.
 Kaplan, L. M.; Resnikoff, M. (1967). "Matrix products and explicit 3, 6, 9, and 12j coefficients of the regular representation of SU(n)". J. Math. Phys. 8 (11): 2194. Bibcode:1967JMP.....8.2194K. doi:10.1063/1.1705141.
 Kaeding, Thomas (1995). "Tables of SU(3) isoscalar factors". Atomic Data and Nuclear Data Tables. 61 (2): 233–288. arXiv:nuclth/9502037. Bibcode:1995ADNDT..61..233K. doi:10.1006/adnd.1995.1011.
 Merzbacher, Eugen (1998). Quantum Mechanics (3rd ed.). John Wiley. pp. 428–9. ISBN 9780471887027.
 Albert Messiah (1966). Quantum Mechanics (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
 de Swart, J. J. (1963). "The Octet model and its ClebschGordan coefficients". Rev. Mod. Phys. (Submitted manuscript). 35 (4): 916. Bibcode:1963RvMP...35..916D. doi:10.1103/RevModPhys.35.916.
External linksEdit
 Nakamura, Kenzo; et al. (2010). "Review of Particle Physics: ClebschGordan coefficients, spherical harmonics, and d functions" (PDF). Journal of Physics G: Nuclear and Particle Physics. 37 (75021): 368.
Partial update for 2012 edition
 Clebsch–Gordan, 3j and 6j Coefficient Web Calculator
 Downloadable Clebsch–Gordan Coefficient Calculator for Mac and Windows
 Web interface for tabulating SU(N) Clebsch–Gordan coefficients
Further readingEdit
 Quantum mechanics, E. Zaarur, Y. Peleg, R. Pnini, Schaum's Easy Oulines Crash Course, McGraw Hill (USA), 2006, ISBN 9780071455336
 Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition), R. Eisberg, R. Resnick, John Wiley & Sons, 1985, ISBN 9780471873730
 Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 9780131461000
 Physics of Atoms and Molecules, B. H. Bransden, C. J. Joachain, Longman, 1983, ISBN 0582444012
 The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 9780521575072.
 Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3527269541, ISBN (VHC Inc.) 0895737523
 McGraw Hill Encyclopaedia of Physics (2nd Edition), C. B. Parker, 1994, ISBN 0070514003
 Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading, Massachusetts: AddisonWesley. ISBN 9780201135077.
 Brink, D. M.; Satchler, G. R. (1993). "Ch. 2". Angular Momentum (3rd ed.). Oxford: Clarendon Press. ISBN 9780198517597.
 Messiah, Albert (1981). "Ch. XIII". Quantum Mechanics (Volume II). New York: North Holland Publishing. ISBN 9780720400458.
 Zare, Richard N. (1988). "Ch. 2". Angular Momentum. New York: John Wiley & Sons. ISBN 9780471858928.