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I'm trying to work out the combination of $|1\ 0 \rangle|0\ 0 \rangle$ (in this case they represent isospin, $|I\ I_3 \rangle$) using Clebsch-Gordan coefficients, but the table for $j_1\times j_2=1\times0$ doesn't appear in the table of Clebsch-Gordan coefficients, they always start at $\frac{1}{2} \times \frac{1}{2}$. Does this mean this combination is not possible? or is there a way to calculate the coefficient?

Thanks in advance

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The spin 0 doesn't do much when added to another spin.

Physicist's "$1/2\otimes 1/2= 1\oplus0$" is $2 \otimes 2=3\oplus 1$ in representation theory language. And "$1\otimes 0$" is simply $3\otimes 1= 3$.

Clebsch-Gordan coefficient: $\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,M\rangle $ is just 1.

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If you take the tensor product of a representation with $j_1 = 1$ and a representation with $j_2 = 0 $, the resulting representation is isomorphic to the $j_1$ representation. This comes about because the $j_2 = 0$ representation, also called the scalar representation, can be just multiplied to the vector. In formulas: pick a vector $v$ and a scalar $s$, then we can identify $$ s \otimes v$$ with $$ s v \ . $$

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