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 $m$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_\pm |jm\rangle = C_\pm(j,m) |j m\pm 1\rangle }

with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle j_1 m_1 | j_2 m_2 \rangle = \delta_{j_1,j_2}\delta_{m_1,m_2}. }

Tensor product space

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_1} be the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2j_1+1} dimensional vector space spanned by the states

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_1 m_1\rangle,\quad m_1=-j_1,-j_1+1,\ldots j_1 }

and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_2} the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2j_2+1} dimensional vector space spanned by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_2 m_2\rangle,\quad m_2=-j_2,-j_2+1,\ldots j_2. }

The tensor product of these spaces, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{12}\equiv V_1\otimes V_2} , has a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2j_1+1)(2j_2+1)} dimensional uncoupled basis

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{12}} can be defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (j_i \otimes 1)|j_1 m_1\rangle|j_2 m_2\rangle \equiv (j_i|j_1m_1\rangle) \otimes |j_2m_2\rangle }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1 \otimes j_i) |j_1 m_1\rangle|j_2 m_2\rangle) \equiv |j_1m_1\rangle \otimes j_i|j_2m_2\rangle. }

Total angular momentum operators are defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [J_k,J_l] = i \sum_{m=1}^3 \epsilon_{klm}J_m }

and hence total angular momentum eigenstates exist

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{J}^2 |(j_1j_2)JM\rangle = J(J+1) |(j_1j_2)JM\rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_z |(j_1j_2)JM\rangle = M |(j_1j_2)JM\rangle,\quad \mathrm{for}\quad M=-J,\ldots,J }

It can be derived [see, e.g., Messiah (1981) pp. 556-558] that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} must satisfy the triangular condition

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{12}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{12}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle J_1 M_1 | J_2 M_2 \rangle = \delta_{J_1J_2}\delta_{M_1M_2} }

Formal definition of Clebsch-Gordan coefficients

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |(j_1j_2)JM\rangle = \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} |j_1m_1\rangle|j_2m_2\rangle \langle j_1m_1j_2m_2|JM\rangle }

The expansion coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle j_1m_1j_2m_2|JM\rangle} are called Clebsch-Gordan coefficients.

Applying the operator

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

• Wigner, E. P. (1931). Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren. Braunschweig: Vieweg Verlag. 
• 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.