Clebsch-Gordan coefficients

In quantum mechanics, the Clebsch-Gordan coefficients (CG coefficients) are sets of numbers that arise in angular momentum coupling.

In mathematics, the CG coefficients appear in group representation theory, particularly of compact Lie groups. They arise in the explicit direct sum decomposition of the outer product of two irreducible representations (irreps) of a group G. In general the outer product representation (rep)—which is carried by a tensor product space—is reducible under G. Decomposition of the outer product rep into irreps of G requires a basis transformation of the tensor product space. The CG coefficients are the elements of the matrix of this basis transformation. In physics it is common to consider only orthonormal bases of the vector spaces involved, and then CG coefficients constitute a unitary matrix.

The name derives from the German mathematicians Alfred Clebsch (1833-1872) and Paul Gordan (1837-1912), who encountered an equivalent problem in invariant theory.

The formulas below use Dirac's bra-ket notation, i.e., the quantity $\langle \psi |\phi \rangle$ stands for a positive definite inner product between the elements ψ and φ of the same complex inner product space. We follow the physical convention $\langle c\psi |\phi \rangle =c^{*}\langle \psi |\phi \rangle$, where $c^{*}\,$ is the complex conjugate of the complex number c.

Clebsch-Gordan coefficients

Although Clebsch-Gordan coefficients can be defined for arbitrary groups, we restrict our attention in this article to the groups associated with space and spin angular momentum, namely the groups SO(3) and SU(2). In that case CG coefficients can be defined as 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 recursion relations for the Clebsch-Gordan coefficients can be found. In order to settle the numerical values for the coefficients, a phase convention must be adopted. Below the Condon and Shortley phase convention is chosen.

Angular momentum operators

Angular momentum operators are Hermitian operators j₁, j₂, and j₃,that satisfy the commutation relations

$[j_{k},j_{l}]=i\sum _{m=1}^{3}\varepsilon _{klm}j_{m},$

where $\varepsilon _{klm}$ is the Levi-Civita symbol. Together the three components define a vector operator $\mathbf {j}$. The square of the length of $\mathbf {j}$ is defined as

$\mathbf {j} ^{2}=j_{1}^{2}+j_{2}^{2}+j_{3}^{2}.$

We also define raising $(j_{+})$ and lowering $(j_{-})$ operators

$j_{\pm }=j_{1}\pm ij_{2}.\,$

Angular momentum states

It can be shown from the above definitions that $\mathbf {j} ^{2}$ commutes with $j_{1},j_{2}$ and $j_{3}$

$[\mathbf {j} ^{2},j_{k}]=0\ \mathrm {for} \ k=1,2,3$

When two Hermitian operators commute a common set of eigenfunctions exists. Conventionally $\mathbf {j} ^{2}$ and $j_{3}$ are chosen. From the commutation relations the possible eigenvalues can be found. The result is

$\mathbf {j} ^{2}|jm\rangle =j(j+1)|jm\rangle ,\qquad j=0,1/2,1,3/2,2,\ldots$

$j_{3}|jm\rangle =m|jm\rangle ,\qquad \quad m=-j,-j+1,\ldots ,j.$

The raising and lowering operators change the value of $m$

$j_{\pm }|jm\rangle =C_{\pm }(j,m)|jm\pm 1\rangle$

with

$C_{\pm }(j,m)={\sqrt {j(j+1)-m(m\pm 1)}}={\sqrt {(j\mp m)(j\pm m+1)}}.$

A (complex) phase factor could be included in the definition of $C_{\pm }(j,m)$ 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 they are assumed to be normalized

$\langle j_{1}m_{1}|j_{2}m_{2}\rangle =\delta _{j_{1},j_{2}}\delta _{m_{1},m_{2}}.$

Tensor product space

Let $V_{1}$ be the $2j_{1}+1$ dimensional vector space spanned by the states

$|j_{1}m_{1}\rangle ,\quad m_{1}=-j_{1},-j_{1}+1,\ldots j_{1}$

and $V_{2}$ the $2j_{2}+1$ dimensional vector space spanned by

$|j_{2}m_{2}\rangle ,\quad m_{2}=-j_{2},-j_{2}+1,\ldots j_{2}.$

The tensor product of these spaces, $V_{12}\equiv V_{1}\otimes V_{2}$, has a $(2j_{1}+1)(2j_{2}+1)$ dimensional uncoupled basis

$|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \equiv |j_{1}m_{1}\rangle \otimes |j_{2}m_{2}\rangle ,\quad m_{1}=-j_{1},\ldots j_{1},\quad m_{2}=-j_{2},\ldots j_{2}.$

Angular momentum operators acting on $V_{12}$ can be defined by

$(j_{i}\otimes 1)|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \equiv (j_{i}|j_{1}m_{1}\rangle )\otimes |j_{2}m_{2}\rangle$

and

$(1\otimes j_{i})|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle )\equiv |j_{1}m_{1}\rangle \otimes j_{i}|j_{2}m_{2}\rangle .$

Total angular momentum operators are defined by

$J_{i}=j_{i}\otimes 1+1\otimes j_{i}\quad \mathrm {for} \quad i=1,2,3$

The total angular momentum operators satisfy the required commutation relations

$[J_{k},J_{l}]=i\sum _{m=1}^{3}\epsilon _{klm}J_{m}$

and hence total angular momentum eigenstates exist

$\mathbf {J} ^{2}|(j_{1}j_{2})JM\rangle =J(J+1)|(j_{1}j_{2})JM\rangle$

$J_{z}|(j_{1}j_{2})JM\rangle =M|(j_{1}j_{2})JM\rangle ,\quad \mathrm {for} \quad M=-J,\ldots ,J$

It can be derived [see, e.g., Messiah (1981) pp. 556-558] that $J$ must satisfy the triangular condition

$|j_{1}-j_{2}|\leq J\leq j_{1}+j_{2}$

The total number of total angular momentum eigenstates is equal to the dimension of $V_{12}$

$\sum _{J=|j_{1}-j_{2}|}^{j_{1}+j_{2}}(2J+1)=(2j_{1}+1)(2j_{2}+1)$

The total angular momentum states form an orthonormal basis of $V_{12}$

$\langle J_{1}M_{1}|J_{2}M_{2}\rangle =\delta _{J_{1}J_{2}}\delta _{M_{1}M_{2}}$

Formal definition of Clebsch-Gordan coefficients

The total angular momentum states can be expanded in the uncoupled basis

$|(j_{1}j_{2})JM\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$

The expansion coefficients $\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$ are called Clebsch-Gordan coefficients.

Applying the operator

$J_{3}=j_{3}\otimes 1+1\otimes j_{3}$

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

$M=m_{1}+m_{2}.\,$

Recursion relations

Applying the total angular momentum raising and lowering operators

$J_{\pm }=j_{\pm }\otimes 1+1\otimes j_{\pm }$

to the left hand side of the defining equation gives

$J_{\pm }|(j_{1}j_{2})JM\rangle =C_{\pm }(J,M)|(j_{1}j_{2})JM\pm 1\rangle =C_{\pm }(J,M)\sum _{m_{1}m_{2}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|JM\pm 1\rangle .$

Applying the same operators to the right hand side gives

$J_{\pm }\sum _{m_{1}m_{2}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$

$=\sum _{m_{1}m_{2}}\left[C_{\pm }(j_{1},m_{1})|j_{1}m_{1}\pm 1\rangle |j_{2}m_{2}\rangle +C_{\pm }(j_{2},m_{2})|j_{1}m_{1}\rangle |j_{2}m_{2}\pm 1\rangle \right]\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$

$=\sum _{m_{1}m_{2}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \left[C_{\pm }(j_{1},m_{1}\mp 1)\langle j_{1}{m_{1}\mp 1}j_{2}m_{2}|JM\rangle +C_{\pm }(j_{2},m_{2}\mp 1)\langle j_{1}m_{1}j_{2}{m_{2}\mp 1}|JM\rangle \right].$

Combining these results gives recursion relations for the Clebsch-Gordan coefficients

$C_{\pm }(J,M)\langle j_{1}m_{1}j_{2}m_{2}|JM\pm 1\rangle =C_{\pm }(j_{1},m_{1}\mp 1)\langle j_{1}{m_{1}\mp 1}j_{2}m_{2}|JM\rangle +C_{\pm }(j_{2},m_{2}\mp 1)\langle j_{1}m_{1}j_{2}{m_{2}\mp 1}|JM\rangle .$

Taking the upper sign with $M=J$ gives

$0=C_{+}(j_{1},m_{1}-1)\langle j_{1}{m_{1}-1}j_{2}m_{2}|JJ\rangle +C_{+}(j_{2},m_{2}-1)\langle j_{1}m_{1}j_{2}m_{2}-1|JJ\rangle .$

In the Condon and Shortley phase convention the coefficient $\langle j_{1}j_{1}j_{2}J-j_{1}|JJ\rangle$ is taken real and positive. With the last equation all other Clebsch-Gordan coefficients $\langle j_{1}m_{1}j_{2}m_{2}|JJ\rangle$ can be found. The normalization is fixed by the requirement that the sum of the squares, which corresponds to the norm of the state $|(j_{1}j_{2})JJ\rangle$ 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 and Shortley phase convention).

Explicit expression

The first derivation of an algebraic formula for CG coefficients was given by Wigner in his famous 1931 book. The following expression for the CG coefficients is due to Van der Waerden and is the most symmetric one of the various existing forms, see, e.g., Biedenharn and Louck (1981) for a derivation,

$\langle jm|j_{1}m_{1};j_{2}m_{2}\rangle =\delta _{m,m_{1}+m_{2}}\Delta (j_{1},j_{2},j)$

$\times \sum _{t}(-1)^{t}{\textstyle {\frac {\left[(2j+1)(j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!(j+m)!(j-m)!\right]^{\frac {1}{2}}}{t!(j_{1}+j_{2}-j-t)!(j_{1}-m_{1}-t)!(j_{2}+m_{2}-t)!(j-j_{2}+m_{1}+t)!(j-j_{1}-m_{2}+t)!}}}$

where

$\Delta (j_{1},j_{2},j)\equiv \left[{\frac {(j_{1}+j_{2}-j)!(j_{1}-j_{2}+j)!(-j_{1}+j_{2}+j)!}{(j_{1}+j_{2}+j+1)!}}\right]^{\frac {1}{2}},$

and the sum runs over all values of t which do not lead to negative factorials.

Orthogonality relations

These are most clearly written down by introducing the alternative notation

$\langle JM|j_{1}m_{1}j_{2}m_{2}\rangle \equiv \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$

The first orthogonality relation is

$\sum _{J=|j_{1}-j_{2}|}^{j_{1}+j_{2}}\sum _{M=-J}^{J}\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle \langle JM|j_{1}m_{1}'j_{2}m_{2}'\rangle =\delta _{m_{1},m_{1}'}\delta _{m_{2},m_{2}'}$

and the second

$\sum _{m_{1}m_{2}}\langle JM|j_{1}m_{1}j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|J'M'\rangle =\delta _{J,J'}\delta _{M,M'}.$

Special cases

For $J=0$ the Clebsch-Gordan coefficients are given by

$\langle j_{1}m_{1}j_{2}m_{2}|00\rangle =\delta _{j_{1},j_{2}}\delta _{m_{1},-m_{2}}{\frac {(-1)^{j_{1}-m_{1}}}{\sqrt {2j_{2}+1}}}.$

For $J=j_{1}+j_{2}$ and $M=J$ we have

$\langle j_{1}j_{1}j_{2}j_{2}|(j_{1}+j_{2})(j_{1}+j_{2})\rangle =1.$

Symmetry properties

$\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle =(-1)^{j_{1}+j_{2}-J}\langle j_{1}{-m_{1}}j_{2}{-m_{2}}|J{-M}\rangle =(-1)^{j_{1}+j_{2}-J}\langle j_{2}m_{2}j_{1}m_{1}|JM\rangle .$

Relation to 3-jm symbols

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

$\langle j_{1}m_{1}j_{2}m_{2}|j_{3}m_{3}\rangle =(-1)^{j_{1}-j_{2}+m_{3}}{\sqrt {2j_{3}+1}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&-m_{3}\end{pmatrix}}.$

See also

• 3-jm symbol
• Racah W-coefficient
• 6-j symbol
• 9-j symbol
• Spherical harmonics
• Associated Legendre polynomials
• Angular momentum
• Angular momentum coupling
• Total electronic angular momentum quantum number
• Azimuthal quantum number
• Table of Clebsch-Gordan coefficients

External links

• Java^TM Clebsch-Gordan Coefficient Calculator
• Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator

References

• Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton, New Jersey: Princeton University Press. ISBN 0-691-07912-9. 

• Condon, Edward U.; Shortley, G. H. (1970). “Chapter 3”, The Theory of Atomic Spectra. Cambridge: Cambridge University Press. ISBN 0-521-09209-4. 

• Messiah, Albert (1981). Quantum Mechanics (Volume II), 12th edition. New York: North Holland Publishing. ISBN 0-7204-0045-7. 

• Brink, D. M.; Satchler, G. R. (1993). “Chapter 2”, Angular Momentum, 3rd edition. Oxford: Clarendon Press. ISBN 0-19-851759-9. 

• Zare, Richard N. (1988). “Chapter 2”, Angular Momentum. New York: John Wiley & Sons. ISBN 0-471-85892-7. 

• Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading, Massachusetts: Addison-Wesley. ISBN 0201135078.