Clebsch–Gordan coefficients

Janssen, T.

doi:10.1107/97809553602060000629

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 52-53

Section 1.2.4.3. Clebsch–Gordan coefficients

T. Janssen^a ^*

^a Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

1.2.4.3. Clebsch–Gordan coefficients

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The tensor product of two irreducible representations of a group K is, in general, reducible. If $[{\bf a}_{i}]$ is a basis for the irreducible representation $[\Gamma_{\alpha}]$ ( $[i=1,\ldots, d_{\alpha}]$ ) and $[{\bf b}_{j}]$ one for $[\Gamma_{\beta}]$ ( $[j=1,\ldots, d_{\beta}]$ ), a basis for the tensor product space is given by $[{\bf e}_{ij} = {\bf a}_{i}\otimes{\bf b}_{j}.]$ On this basis, the matrix representation is, in general, not in reduced form, even if the product representation is reducible. Suppose that $[\Gamma_{\alpha}\otimes\Gamma_{\beta} \sim \textstyle\sum\limits_{\gamma} ^{}{}^{\oplus} m_{\gamma}\Gamma_{\gamma}.]$ This means that there is a basis $[\psi_{\gamma\ell k} \quad (\ell =1,\ldots, m_{\gamma};\,k=1,\ldots, d_{\gamma}),]$ on which the representation is in reduced form. The multiplicity $[m_{\gamma}]$ gives the number of times the irreducible component $[\Gamma_{\gamma}]$ occurs in the tensor product. The basis transformation is given by $[\psi_{\gamma \ell k}= \sum\limits_{ij}\left.\left(\matrix{ \alpha & \beta\cr i & j}\right|\matrix{\gamma & \cr k & \ell}\right){\bf a}_{i}\otimes{\bf b}_{j}. \eqno (1.2.4.8)]$ The basis transformation is unitary if one starts with orthonormal bases and has coefficients $[\left.\left(\matrix{ \alpha & \beta\cr i & j}\right|\matrix{\gamma & \cr k & \ell}\right)\eqno (1.2.4.9)]$ called Clebsch–Gordan coefficients. For the group O(3) they are the original Clebsch–Gordan coefficients; for bases $[|\ell m\rangle]$ and $[|\ell' m'\rangle]$ of the ( $[2\ell+1]$ )- and ( $[2\ell'+1]$ )-dimensional representations $[D_{\ell}]$ and $[D_{\ell'}]$ , respectively, of O(3) one has $[\eqalignno{&|JM\rangle = \sum\limits_{m\,m'}\left.\left(\matrix{ \ell & \ell'\cr m & m'}\right|\matrix{J\cr M}\right) |\ell m\rangle\otimes |\ell'm'\rangle,&\cr&\quad(J=|\ell-\ell'|,\ldots, \ell+\ell').&(1.2.4.10)}]$ The multiplicity here is always zero or unity, which is the reason why one leaves out the number $[\ell]$ in the notation.

If the multiplicity $[m_{\gamma}]$ is unity, the coefficients for given $[\alpha, \beta, \gamma]$ are unique up to a common factor for all [i,j,k] . This is no longer the case if the multiplicity is larger, because then one can make linear combinations of the basis vectors belonging to $[\Gamma_{\gamma}]$ . Anyway, one has to follow certain conventions. In the case of O(3), for example, there are the Condon–Shortley phase conventions. The degree of freedom of the Clebsch–Gordan coefficients for given matrix representations $[\Gamma_{\alpha}]$ can be seen as follows. Suppose that there are two basis transformations, S and [S'] , in the tensor product space which give the same reduced form: $[S\left(D_{\alpha}\otimes D_{\beta} \right)S^{-1} = S'\left(D_{\alpha}\otimes D_{\beta} \right)S'^{-1} = D = \bigoplus m_{\gamma}D_{\gamma}. \eqno (1.2.4.11)]$ Then the matrix $[S'S^{-1}]$ commutes with every matrix [D(R)] ( $[R\in K]$ ). If all multiplicities are zero or unity, it follows from Schur's lemma that $[S'S^{-1}]$ is the direct sum of unit matrices of dimension $[d_{\gamma}]$ . If the multiplicities are larger, the matrix $[S'S^{-1}]$ is a direct sum of blocks which are of the form $[\pmatrix{ \lambda_{11}E & \lambda_{12}E & \ldots &\lambda_{1m_{\gamma}}E \cr \lambda_{21}E & \lambda_{22}E & \ldots &\lambda_{2m_{\gamma}}E \cr \vdots & \vdots & \ddots & \vdots \cr \lambda_{m_{\gamma}1}E & \ldots & \ldots & \lambda_{m_{\gamma}m_{\gamma}}E }, ]$ such that $[{\rm Det}(\lambda_{ij})=1]$ , and the E's are $[d_{\gamma}]$ -dimensional unit matrices. This means that for multiplicity-free ( $[m_{\gamma} \,\leq \, 1]$ ) cases, the Clebsch–Gordan coefficients are unique up to a common factor for all coefficients involving one value of $[\gamma]$ .

The Clebsch–Gordan coefficients satisfy the following rules: $[\displaylines{\left.\left(\matrix{ \alpha & \beta\cr i & j}\right|\matrix{\gamma & \cr k & \ell}\right) = \left.\left(\matrix{ \beta & \alpha\cr j & i}\right|\matrix{\gamma & \cr k & \ell}\right)\cr\left.\left(\matrix{ \alpha & \beta\cr i & j}\right|\matrix{\gamma & \cr k & \ell}\right) = 0, \hbox{ if } D_\alpha\otimes D_\beta \hbox{ does not contain } D_\gamma \cr \sum\limits_{k\ell}\left.\left(\matrix{ \alpha & \beta\cr i & j}\right|\matrix{\gamma & \cr k & \ell}\right)^* \left.\left(\matrix{ \alpha & \beta\cr i' & j'}\right|\matrix{\gamma & \cr k & \ell}\right) = \delta_{ii'}\delta_{jj'} \cr \sum\limits_{ij}\left.\left(\matrix{ \alpha & \beta\cr i & j}\right|\matrix{\gamma & \cr k & \ell}\right)^* \left.\left(\matrix{ \alpha & \beta\cr i & j}\right|\matrix{\gamma & \cr k' & \ell'}\right) = \delta_{kk'}\delta_{\ell \ell '}.}]$ For the basis vectors of the invariant space belonging to the identity representation $[\Gamma_{1}]$ , one has $[\gamma =d_{\gamma}=1]$ . Consequently, $[\psi_{\ell} = \sum\limits_{ij}\left.\left(\matrix{ \alpha & \beta\cr i & j}\right|\matrix{1 & \cr 1 & \ell}\right) {\bf a}_{i}\otimes {\bf b}_{j}.]$

References

International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 52-53