I have a general question on finding momentum states

Let's say I have two spin 1/2 particles, so
$$J=S_1+S_2$$
and |J| ranges from $$|S_1 + S_2|$$ to $$|S_1 - S_2|$$
in this case J=1 is triplet and J=0 is singlet.

Now how do you find the J=0 state? I know that
$$|j=0,m_j=0>=\frac{1}{\sqrt{2}}(\uparrow_1 \downarrow_2 - \downarrow_1 \uparrow_2)$$
but how do you get this in the first place? Is it pretty much trial and error and then use operator to generate rest of the states for that particular J? Or is there an algorithm for finding state $$|j,m_j=j>$$?

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dextercioby
Homework Helper
Well, it's all about reading & understanding Clebsch-Gordan theorem & coefficients correctly. Technically

$$|j,m\rangle =\sum_{m_{1},m_{2}} \langle j_{1},m_{1},j_{2},m_{2}|j,m\rangle |j_{1},m_{1},j_{2},m_{2} \rangle$$

The coefficients are tabulated, also, you know that j=0 and m=0.

Daniel.

Meir Achuz