Racah W-coefficient

Coupling of two angular momenta ${\displaystyle \mathbf {j} _{1}}$ and ${\displaystyle \mathbf {j} _{2}}$ is the construction of simultaneous eigenfunctions of ${\displaystyle \mathbf {J} ^{2}}$ and ${\displaystyle J_{z}}$, where ${\displaystyle \mathbf {J} =\mathbf {j} _{1}+\mathbf {j} _{2}}$, as explained in the article on Clebsch–Gordan coefficients. The result is

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

where ${\displaystyle J=|j_{1}-j_{2}|,\ldots ,j_{1}+j_{2}}$ and ${\displaystyle M=-J,\ldots ,J}$.

Coupling of three angular momenta ${\displaystyle \mathbf {j} _{1}}$, ${\displaystyle \mathbf {j} _{2}}$, and ${\displaystyle \mathbf {j} _{3}}$, may be done by first coupling ${\displaystyle \mathbf {j} _{1}}$ and ${\displaystyle \mathbf {j} _{2}}$ to ${\displaystyle \mathbf {J} _{12}}$ and next coupling ${\displaystyle \mathbf {J} _{12}}$ and ${\displaystyle \mathbf {j} _{3}}$ to total angular momentum ${\displaystyle \mathbf {J} }$:

${\displaystyle |((j_{1}j_{2})J_{12}j_{3})JM\rangle =\sum _{M_{12}=-J_{12}}^{J_{12}}\sum _{m_{3}=-j_{3}}^{j_{3}}|(j_{1}j_{2})J_{12}M_{12}\rangle |j_{3}m_{3}\rangle \langle J_{12}M_{12}j_{3}m_{3}|JM\rangle }$

Alternatively, one may first couple ${\displaystyle \mathbf {j} _{2}}$ and ${\displaystyle \mathbf {j} _{3}}$ to ${\displaystyle \mathbf {J} _{23}}$ and next couple ${\displaystyle \mathbf {j} _{1}}$ and ${\displaystyle \mathbf {J} _{23}}$ to ${\displaystyle \mathbf {J} }$:

${\displaystyle |(j_{1},(j_{2}j_{3})J_{23})JM\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{M_{23}=-J_{23}}^{J_{23}}|j_{1}m_{1}\rangle |(j_{2}j_{3})J_{23}M_{23}\rangle \langle j_{1}m_{1}J_{23}M_{23}|JM\rangle }$

Both coupling schemes result in complete orthonormal bases for the ${\displaystyle (2j_{1}+1)(2j_{2}+1)(2j_{3}+1)}$ dimensional space spanned by

${\displaystyle |j_{1}m_{1}\rangle |j_{2}m_{2}\rangle |j_{3}m_{3}\rangle ,\;\;m_{1}=-j_{1},\ldots ,j_{1};\;\;m_{2}=-j_{2},\ldots ,j_{2};\;\;m_{3}=-j_{3},\ldots ,j_{3}.}$

Hence, the two total angular momentum bases are related by a unitary transformation. The matrix elements of this unitary transformation are given by a scalar product and are known as recoupling coefficients. The coefficients are independent of ${\displaystyle M}$ and so we have

${\displaystyle |((j_{1}j_{2})J_{12}j_{3})JM\rangle =\sum _{J_{23}}|(j_{1},(j_{2}j_{3})J_{23})JM\rangle \langle (j_{1},(j_{2}j_{3})J_{23})J|((j_{1}j_{2})J_{12}j_{3})J\rangle .}$

The independence of ${\displaystyle M}$ follows readily by writing this equation for ${\displaystyle M=J}$ and applying the lowering operator ${\displaystyle J_{-}}$ to both sides of the equation.

Let

${\displaystyle \Delta (a,b,c)=[(a+b-c)!(a-b+c)!(-a+b+c)!/(a+b+c+1)!]^{1/2}}$

be the usual triangular factor, then the Racah coefficient is a product of four of these by a sum over factorials,

${\displaystyle W(abcd;ef)=\Delta (a,b,e)\Delta (c,d,e)\Delta (a,c,f)\Delta (b,d,f)w(abcd;ef)}$

where

${\displaystyle w(abcd;ef)\equiv \sum _{z}{\frac {(-1)^{z+\beta _{1}}(z+1)!}{(z-\alpha _{1})!(z-\alpha _{2})!(z-\alpha _{3})!(z-\alpha _{4})!(\beta _{1}-z)!(\beta _{2}-z)!(\beta _{3}-z)!}}}$

and

${\displaystyle \alpha _{1}=a+b+e;\quad \beta _{1}=a+b+c+d;}$

${\displaystyle \alpha _{2}=c+d+e;\quad \beta _{2}=a+d+e+f;}$

${\displaystyle \alpha _{3}=a+c+f;\quad \beta _{3}=b+c+e+f;}$

${\displaystyle \alpha _{4}=b+d+f.}$

The sum over ${\displaystyle z}$ is finite over the range[3]

${\displaystyle \max(\alpha _{1},\alpha _{2},\alpha _{3},\alpha _{4})\leq z\leq \min(\beta _{1},\beta _{2},\beta _{3}).}$

Racah's W-coefficients are related to Wigner's 6-j symbols, which have even more convenient symmetry properties

${\displaystyle W(abcd;ef)(-1)^{a+b+c+d}={\begin{Bmatrix}a&b&e\\d&c&f\end{Bmatrix}}.}$

Cf.[4] or

${\displaystyle W(j_{1}j_{2}Jj_{3};J_{12}J_{23})=(-1)^{j_{1}+j_{2}+j_{3}+J}{\begin{Bmatrix}j_{1}&j_{2}&J_{12}\\j_{3}&J&J_{23}\end{Bmatrix}}.}$

• Clebsch–Gordan coefficients
• 3-j symbol
• 6-j symbol
• Pandya theorem

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