Clebsch–Gordan coefficients explained

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 quantum-mechanical 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. 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.

The formulas below use Dirac's bra–ket notation and the Condon–Shortley phase convention is adopted.

Review of the angular momentum operators

Angular momentum operators are self-adjoint operators,, and that satisfy the commutation relationswhere is the Levi-Civita symbol. Together the three operators define a vector operator, a rank one Cartesian tensor operator,$$\backslash mathbf\; j\; =\; (\backslash mathrm,\; \backslash mathrm,\; \backslash mathrm).$$It is 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 as the inner product of with itself:$$\backslash mathbf\; j^2\; =\; \backslash mathrm\; +\; \backslash mathrm\; +\; \backslash mathrm.$$This is an example of a Casimir operator. It is diagonal and its eigenvalue characterizes the particular irreducible representation of the angular momentum algebra

ak{so}(3, R ) \cong ak{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 and lowering operators, the so-called ladder operators,$$\backslash mathrm\; =\; \backslash mathrm\; \backslash pm\; i\; \backslash mathrm.$$

Spherical basis for angular momentum eigenstates

It can be shown from the above definitions that commutes with,, and :

When two Hermitian operators commute, a common set of eigenstates exists. Conventionally, and are chosen. From the commutation relations, the possible eigenvalues can be found. These eigenstates are denoted where is the angular momentum quantum number and is the angular momentum projection onto the z-axis.

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,$$\backslash mathrm\; j\_\backslash pm\; |j\; \backslash ,\; m\backslash rangle\; =\; \backslash hbar\; C\_\backslash pm(j,\; m)\; |j\; \backslash ,\; (m\; \backslash pm\; 1)\backslash rangle,$$where the ladder coefficient is given by:

In principle, one may also introduce a (possibly complex) phase factor in the definition of

C _{\pm(j,} m) . 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,$$\backslash langle\; j\; \backslash ,\; m\; |\; j\text{'}\; \backslash ,\; m\text{'}\; \backslash rangle\; =\; \backslash delta\_\; \backslash delta\_.$$

Here the italicized and denote integer or half-integer angular momentum quantum numbers of a particle or of a system. On the other hand, the roman,,,,, and denote operators. The

\delta symbols are Kronecker deltas.

Tensor product space

We now consider systems with two physically different angular momenta and . 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

V _{1} of dimension 2 j _{1+1} and also on a space V _{2} of dimension 2 j _{2} + 1 . We are then going to define a family of "total angular momentum" operators acting on the tensor product space V _{1} ⊗ V _{2} , which has dimension ( 2 j _{1+1)(2j} _{2+1)} . 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 be the -dimensional vector space spanned by the statesand the -dimensional vector space spanned by the states

The tensor product of these spaces,, has a -dimensional uncoupled basis$$|j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\backslash rangle\; \backslash equiv\; |j\_1\; \backslash ,\; m\_1\backslash rangle\; \backslash otimes\; |j\_2\; \backslash ,\; m\_2\backslash rangle,\; \backslash quad\; m\_1\; \backslash in\; \backslash ,\; \backslash quad\; m\_2\; \backslash in\; \backslash .$$Angular momentum operators are defined to act on states in in the following manner:$$(\backslash mathbf\; j\; \backslash otimes\; 1)\; |j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\backslash rangle\; \backslash equiv\; \backslash mathbf\; j\; |j\_1\; \backslash ,\; m\_1\backslash rangle\; \backslash otimes\; |j\_2\; \backslash ,\; m\_2\backslash rangle$$and$$(1\; \backslash otimes\; \backslash mathrm\; \backslash mathbf\; j)\; |j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\backslash rangle\; \backslash equiv\; |j\_1\; \backslash ,\; m\_1\backslash rangle\; \backslash otimes\; \backslash mathbf\; j\; |j\_2\; \backslash ,\; m\_2\backslash rangle,$$where denotes the identity operator.

The total^[1] angular momentum operators are defined by the coproduct (or tensor product) of the two representations acting on,

The total angular momentum operators can be shown to satisfy the very same commutation relations, $$[\backslash mathrm\{J\}\_k,\; \backslash mathrm\{J\}\_l]\; =\; i\; \backslash hbar\; \backslash varepsilon\_\; \backslash mathrm\_m\; ~,$$where . Indeed, the preceding construction is the standard method^[2] 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 . Note that it is common to omit the part.

The total angular momentum quantum number must satisfy the triangular condition that$$|j\_1\; -\; j\_2|\; \backslash leq\; J\; \backslash leq\; j\_1\; +\; j\_2,$$such that the three nonnegative integer or half-integer values could correspond to the three sides of a triangle.

The total number of total angular momentum eigenstates is necessarily equal to the dimension of :$$\backslash sum\_^\; (2\; J\; +\; 1)\; =\; (2\; j\_1\; +\; 1)\; (2\; j\_2\; +\; 1)\; ~.$$As this computation suggests, the tensor product representation decomposes as the direct sum of one copy of each of the irreducible representations of dimension

2 J +1 , where J ranges from |j _{1} - j _{2|} to j _{1} + j _{2} in increments of 1.^[3] As an example, consider the tensor product of the three-dimensional representation corresponding to j _{1} = 1 with the two-dimensional representation with j _{2} = 1 / 2 . The possible values of J are then J = 1 / 2 and J = 3 / 2 . Thus, the six-dimensional tensor product representation decomposes as the direct sum of a two-dimensional representation and a four-dimensional 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 :$$\backslash left\backslash langle\; J\backslash ,\; M\; |\; J\text{'}\backslash ,\; M\text{'}\; \backslash right\backslash rangle\; =\; \backslash delta\_\backslash delta\_~.$$

These rules may be iterated to, e.g., combine doublets (=1/2) to obtain the Clebsch-Gordan decomposition series, (Catalan's triangle),$$\backslash mathbf^\; =\; \backslash bigoplus\_^~\; \backslash left(\backslash frac\backslash right)~(\backslash mathbf\; +\; \backslash mathbf\; -\; \backslash mathbf\backslash mathbf)~,$$where

\lfloor n/ 2 \rfloor is the integer floor function; and the number preceding the boldface irreducible representation dimensionality label indicates multiplicity of that representation in the representation reduction.^[4] For instance, from this formula, addition of three spin 1/2s yields a spin 3/2 and two spin 1/2s, { 2} ⊗ { 2} ⊗ { 2} = { 4} ⊕ { 2} ⊕ { 2} .

Formal definition of Clebsch–Gordan coefficients

The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis

The expansion coefficients

$$\backslash langle\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\; |\; J\; \backslash ,\; M\; \backslash rangle$$

are the Clebsch–Gordan coefficients. Note that some authors write them in a different order such as . Another common notation is.

Applying the operators

to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when

Recursion relations

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$$\backslash mathrm\; J\_\backslash pm\; =\; \backslash mathrm\; j\_\backslash pm\; \backslash otimes\; 1\; +\; 1\; \backslash otimes\; \backslash mathrm\; j\_\backslash pm$$to the left hand side of the defining equation givesApplying the same operators to the right hand side gives

Combining these results gives recursion relations for the Clebsch–Gordan coefficients, where was defined in :$$C\_\backslash pm(J,\; M)\; \backslash langle\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\; |\; J\; \backslash ,\; (M\; \backslash pm\; 1)\backslash rangle\; =\; C\_\backslash pm(j\_1,\; m\_1\; \backslash mp\; 1)\; \backslash langle\; j\_1\; \backslash ,\; (m\_1\; \backslash mp\; 1)\; \backslash ,\; j\_2\; \backslash ,\; m\_2\; |\; J\; \backslash ,\; M\backslash rangle\; +\; C\_\backslash pm(j\_2,\; m\_2\; \backslash mp\; 1)\; \backslash langle\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; (m\_2\; \backslash mp\; 1)\; |\; J\; \backslash ,\; M\backslash rangle.$$

Taking the upper sign with the condition that gives initial recursion relation:$$0\; =\; C\_+(j\_1,\; m\_1\; -\; 1)\; \backslash langle\; j\_1\; \backslash ,\; (m\_1\; -\; 1)\; \backslash ,\; j\_2\; \backslash ,\; m\_2\; |\; J\; \backslash ,\; J\backslash rangle\; +\; C\_+(j\_2,\; m\_2\; -\; 1)\; \backslash langle\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; (m\_2\; -\; 1)\; |\; J\; \backslash ,\; J\backslash rangle.$$In the Condon–Shortley phase convention, one adds the constraint that

\langle j _{1} j _{1} j _{2} (J - j _{1)} | J J\rangle > 0 (and is therefore also real).The Clebsch–Gordan coefficients 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 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–Shortley phase convention.

Orthogonality relations

These are most clearly written down by introducing the alternative notation$$\backslash langle\; J\; \backslash ,\; M\; |\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\; \backslash rangle\; \backslash equiv\; \backslash langle\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\; |\; J\; \backslash ,\; M\; \backslash rangle$$

The first orthogonality relation is$$\backslash sum\_^\; \backslash sum\_^J\; \backslash langle\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\; |\; J\; \backslash ,\; M\; \backslash rangle\; \backslash langle\; J\; \backslash ,\; M\; |\; j\_1\; \backslash ,\; m\_1\text{'}\; \backslash ,\; j\_2\; \backslash ,\; m\_2\text{'}\; \backslash rangle\; =\; \backslash langle\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\; |\; j\_1\; \backslash ,\; m\_1\text{'}\; \backslash ,\; j\_2\; \backslash ,\; m\_2\text{'}\; \backslash rangle\; =\; \backslash delta\_\; \backslash delta\_$$(derived from the fact that $\backslash mathbf\; 1\; =\; \backslash sum\_x\; |x\backslash rangle\; \backslash langle\; x|$) and the second one is$$\backslash sum\_\; \backslash langle\; J\; \backslash ,\; M\; |\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\; \backslash rangle\; \backslash langle\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\; |\; J\text{'}\; \backslash ,\; M\text{'}\; \backslash rangle\; =\; \backslash langle\; J\; \backslash ,\; M\; |\; J\text{'}\; \backslash ,\; M\text{'}\; \backslash rangle\; =\; \backslash delta\_\; \backslash delta\_.$$

Special cases

For the Clebsch–Gordan coefficients are given by$$\backslash langle\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_2\; \backslash ,\; m\_2\; |\; 0\; \backslash ,\; 0\; \backslash rangle\; =\; \backslash delta\_\; \backslash delta\_\; \backslash frac.$$

For and we have$$\backslash langle\; j\_1\; \backslash ,\; j\_1\; \backslash ,\; j\_2\; \backslash ,\; j\_2\; |\; (j\_1\; +\; j\_2)\; \backslash ,\; (j\_1\; +\; j\_2)\; \backslash rangle\; =\; 1.$$

For and we have$$\backslash langle\; j\_1\; \backslash ,\; m\_1\; \backslash ,\; j\_1\; \backslash ,\; (-m\_1)\; |\; (2\; j\_1)\; \backslash ,\; 0\; \backslash rangle\; =\; \backslash frac.$$

For we have$$\backslash langle\; j\_1\; \backslash ,\; j\_1\; \backslash ,\; j\_1\; \backslash ,\; (-j\_1)\; |\; J\; \backslash ,\; 0\; \backslash rangle\; =\; (2\; j\_1)!\; \backslash sqrt.$$

For, we have

For we have

Symmetry properties

A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to Wigner 3-j symbols using . The symmetry properties of Wigner 3-j symbols are much simpler.

Rules for phase factors

Care is needed when simplifying phase factors: a quantum number may be a half-integer rather than an integer, therefore is not necessarily for a given quantum number unless it can be proven to be an integer. Instead, it is replaced by the following weaker rule:$$(-1)^\; =\; 1$$for any angular-momentum-like quantum number .

Nonetheless, a combination of and is always an integer, so the stronger rule applies for these combinations:$$(-1)^\; =\; 1$$This identity also holds if the sign of either or or both is reversed.

It is useful to observe that any phase factor for a given pair can be reduced to the canonical form:$$(-1)^$$where and (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 pairs such as the one described in the next paragraph.)

An additional rule holds for combinations of,, and that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol:$$(-1)^\; =\; 1$$This identity also holds if the sign of any is reversed, or if any of them are substituted with an instead.

Relation to Wigner 3-j symbols

Clebsch–Gordan coefficients are related to Wigner 3-j symbols which have more convenient symmetry relations.

The factor is due to the Condon–Shortley constraint that, while is due to the time-reversed nature of .

This allows to reach the general expression:

\begin{align} \langle j _{1} m _{1} j _{2} m _{2} | J M \rangle & \equiv \delta(m _{1+m} _{2,M)} \sqrt{ (( 2 J +1 )(j _{1+j} _{2-J)!(j} _{1-j} _{2+J)!(-j} _{1+j} _{2+J)!})/((j _{1+j} _{2+J+1)!}) }\ \times \\[6pt]&\times\sqrt\ \times \\[6pt]&\times\sum_^N \frac,\end.

The summation is performed over those integer values for which the argument of each factorial in the denominator is non-negative, i.e. summation limits and are taken equal: the lower one

K =max ( 0 , j _{2-J-m} _{1,} j _{1-J+m} _{2),} the upper one N =min (j _{1+j} _{2-J,} j _{1-m} _{1,} j _{2+m} _{2).} Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3j symbol at, for example, j _{3>j} _{1+j} _{2} or j _{1<m} _{1} are automatically set to zero.

Relation to Wigner D-matrices

See main article: Wigner D-matrix.

Relation to spherical harmonics

In the case where integers are involved, the coefficients can be related to integrals of spherical harmonics:$$\backslash int\_\; Y\_^^*(\backslash Omega)\; Y\_^^*(\backslash Omega)\; Y\_L^M\; (\backslash Omega)\; \backslash ,\; d\; \backslash Omega\; =\; \backslash sqrt\; \backslash langle\; \backslash ell\_1\; \backslash ,\; 0\; \backslash ,\; \backslash ell\_2\; \backslash ,\; 0\; |\; L\; \backslash ,\; 0\; \backslash rangle\; \backslash langle\; \backslash ell\_1\; \backslash ,\; m\_1\; \backslash ,\; \backslash ell\_2\; \backslash ,\; m\_2\; |\; L\; \backslash ,\; M\; \backslash rangle$$

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 of a single spherical harmonic:$$Y\_^(\backslash Omega)\; Y\_^(\backslash Omega)\; =\; \backslash sum\_\; \backslash sqrt\; \backslash langle\; \backslash ell\_1\; \backslash ,\; 0\; \backslash ,\; \backslash ell\_2\; \backslash ,\; 0\; |\; L\; \backslash ,\; 0\; \backslash rangle\; \backslash langle\; \backslash ell\_1\; \backslash ,\; m\_1\; \backslash ,\; \backslash ell\_2\; \backslash ,\; m\_2\; |\; L\; \backslash ,\; M\; \backslash rangle\; Y\_L^M\; (\backslash Omega)$$

Other properties

$$\backslash sum\_m\; (-1)^\; \backslash langle\; j\; \backslash ,\; m\; \backslash ,\; j\; \backslash ,\; (-m)\; |\; J\; \backslash ,\; 0\; \backslash rangle\; =\; \backslash delta\_\; \backslash sqrt$$

Clebsch–Gordan coefficients for specific groups

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 SU(n) are known. In particular, SU(3) Clebsch-Gordan coefficients have been computed and tabulated because of their utility in characterizing hadronic decays, where a flavor-SU(3) symmetry exists that relates the up, down, and strange quarks.^[5] A web interface for tabulating SU(N) Clebsch–Gordan coefficients is readily available.

See also

• 3-j symbol
• 6-j symbol
• 9-j symbol
• Racah W-coefficient
• 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 D-matrix
• Wigner–Eckart theorem
• Angular momentum diagrams (quantum mechanics)
• Clebsch–Gordan coefficient for SU(3)
• Littlewood–Richardson coefficient

References

• A.. Alex. M.. Kalus. A.. Huckleberry. J.. von Delft . A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients. J. Math. Phys.. 82. 2 . 2011. 023507. 10.1063/1.3521562. 2011JMP....52b3507A. 1009.0437. 55572438 .
• Book: Condon. Edward U.. Shortley. G. H.. The Theory of Atomic Spectra. 1970. Cambridge University Press. Cambridge. 978-0-521-09209-8. Ch. 3.
• Book: Edmonds, A. R.. Angular Momentum in Quantum Mechanics. 1957. Princeton University Press. Princeton, New Jersey. 978-0-691-07912-7. registration.
• Book: Greiner. Walter. Walter Greiner. Müller. Berndt. Quantum Mechanics: Symmetries. registration. 2nd. 1994. Springer Verlag. 978-3540580805.
  • L. M.. Kaplan. M.. Resnikoff. Matrix products and explicit 3, 6, 9, and 12j coefficients of the regular representation of SU(n). 1967. 10.1063/1.1705141. J. Math. Phys.. 8. 11 . 2194. 1967JMP.....8.2194K .
• Kaeding. Thomas. nucl-th/9502037. Tables of SU(3) isoscalar factors. 1995 . 10.1006/adnd.1995.1011 . 61. 2. Atomic Data and Nuclear Data Tables. 233–288 . 1995ADNDT..61..233K.
• Book: Merzbacher, Eugen. Eugen Merzbacher. Quantum Mechanics. limited. 3rd. 1998. John Wiley. 428–9. 978-0-471-88702-7.
• Albert Messiah (1966). Quantum Mechanics (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
• J. J.. de Swart. The Octet model and its Clebsch-Gordan coefficients . 1963 . 10.1103/RevModPhys.35.916. Rev. Mod. Phys.. 35. 4. 916 . 1963RvMP...35..916D . Submitted manuscript.

External links

• Nakamura. Kenzo. Hagiwara. Kaoru. 1. 2010. Review of Particle Physics: Clebsch-Gordan coefficients, spherical harmonics, and d functions. Journal of Physics G: Nuclear and Particle Physics. 37. 75021. 368. 10.1088/0954-3899/37/7A/075021 . 2010JPhG...37g5021N . Partial update for 2012 edition.
• 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

Further reading

• Book: E. . Zaarur . Y. . Peleg . R. . Pnini . Quantum mechanics . McGraw Hill . Schaum's Easy Oulines Crash Course . 2006 . 978-007-145533-6.
• Book: 2nd . R. . Eisberg . R. . Resnick . Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles . Wiley . 1985 . 978-0-471-87373-0.
• Book: Abers, E. . Quantum Mechanics . Prentice Hall . 2004 . 978-0-13-146100-0.
• Book: B.H. . Bransden . C.J. . Joachain . Physics of Atoms and Molecules . Longman . 1983 . 0-582-44401-2.
• Book: Woan, G. . The Cambridge Handbook of Physics Formulas . Cambridge University Press . 2010 . 978-0-521-57507-2.
• Book: 2nd . Rita G. Lerner . R.G. . Lerner . G.L. . Trigg . Encyclopaedia of Physics . VHC publishers . 1991 . 0-89573-752-3.
• Book: Parker, C.B. . McGraw Hill Encyclopaedia of Physics . 2nd . 1994 . 0-07-051400-3.
• Book: Biedenharn . L.C. . Louck . J.D. . Angular Momentum in Quantum Physics . 1981 . Addison-Wesley . Reading, Massachusetts . 978-0-201-13507-7 .
• Book: Brink . D.M. . Satchler . G.R. . Angular Momentum . 1993 . 3rd . Clarendon Press . 978-0-19-851759-7 . 2. Representations of the Rotation Group . registration .
• Book: Messiah, Albert . Quantum Mechanics . II . 1981 . North Holland . 978-0-7204-0045-8 . XIII. Angular Momentum in Quantum Mechanics . 507– . .
• Book: Zare, Richard N. . Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics . 1988 . Wiley . 978-0-471-85892-8 . 2. Coupling of Two Angular Momentum Vectors . 43–.

Notes and References

1. 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 and . 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.
2. Section 4.3.2
3. Appendix C
4. Zachos . C K. 1992 . Altering the Symmetry of Wavefunctions in Quantum Algebras and Supersymmetry . Modern Physics Letters A . A7 . 18. 1595–1600 . 10.1142/S0217732392001270. hep-th/9203027 . 1992MPLA....7.1595Z . 16360975 .
5. Web site: Coleman . Sidney . Fun with SU(3) . INSPIREHep.