# How to use Wikipedia's Table of Clebsch–Gordan coefficients?

Wikipedia has a nice article outlining Clebsch-Gordan coefficients.

For example, to my understaning, this table tells us how to combine two particles, each having a maximum total angular momentum $$1$$ into one wavefunction with maximum angular momentum $$2$$:

Take the first column from the last table. It tells us, I believe:

$$|2,0\rangle = \sqrt{\frac{1}{6}} |1,1\rangle |1,-1\rangle +\sqrt{\frac{2}{3}}|1,0\rangle|1,0\rangle+\sqrt{\frac{1}{6}} |1,-1\rangle|1,1\rangle$$

How I interpret this:

The total angular momentum of a particle which arises from such a combination of wavefunctions of two other particles will have total angular quantum number 2 (so total angular momentum $$\sqrt{j(j+1)\hbar^2}=\sqrt{2(2+1)\hbar^2}$$), but $$0$$ around the $$z$$ axis (as $$m_j$$, what I understand to be the angular momentum around the $$z$$ axis, is $$0$$).

So the constituent particle's angular momentum is not aligned with each other, in fact they are antialigned enough so that the total z-directional angular momentum will be 0.

Is this interpretation of what's going on correct? My concern is that there are no tables for $$m=-1,-2$$. If my interpretation of the situation is correct, I see no reason why I couldn't produce a combined particle with these $$m$$ values, if I can do it for $$m=0,1,2$$.

For brevity, solutions with $$M < 0$$ and $$j_1 < j_2$$ are omitted. They may be calculated using the simple relations $$\langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle =(-1)^{J-j_{1}-j_{2}}\langle j_{1},j_{2};-m_{1},-m_{2}\mid j_{1},j_{2};J,-M\rangle .$$ and $$\langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle =(-1)^{J-j_{1}-j_{2}}\langle j_{2},j_{1};m_{2},m_{1}\mid j_{2},j_{1};J,M\rangle.$$
In other words, the Clebsch-Gordon coefficients for a negative value of $$m$$ are the same (up to a sign) as those for the corresponding positive value of $$m$$, so long as you switch the signs of $$m_1$$ and $$m_2$$ as well.