Add. Angular Momentum: Finding j=2 States

Zero86
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Homework Statement


There are two independent subsystems with angular momenta j1 = j2 = 1. States have to be found for the whole system with angular momentum j = 2.

Homework Equations



Basic procedure for addition of Angular Momentum in Quantum Mechanics

The Attempt at a Solution



Basically j = j1 + j2

I go through the states starting with 2 using the equations in the guide.
So basically we are limited to states
|2, 2>
|2, 1>, |1, 1>
|2, 0>, |1, 0>, |0, 0>

But I'm not sure if it's the right way to go through the states. Also, maybe there are better guides since I'm not even sure if this one applies to the problem. I'm looking into Clebsch–Gordan coefficients at the moment.
 
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What do you mean "the right way to go through the states"?
 


Well I have found that there are 2*2+1 = 9 states which are |2, 2>, |2, 1>, |2, 0>, |2, -1>, |2, -2>, |1, 1>, |1, 0>, |1, -1>, |0, 0>.
In 'big O' notation,
1X1 = 2+1+0+|-1|+|-2|
3*3 = 9
So these seem to be all the possible states. I'm not sure if it means I have found all the states. I'm not sure how they have come up with
\ket{j, j-1} = \sqrt{j_1 \over j} \ket{j_1, j_1-1, j_2, j_2} + \sqrt{j_2 \over j} \ket{j_1, j_1, j_2, j_2 - 1}
 


The states with total angular momentum j=2 are |2, 2>, |2, 1>, |2, 0>, |2, -1>, and |2, -2>. You're probably expected to find them in terms of the eigenstates of J1 and J2.
 

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