# Total angular momentum state using two ways

Consider addition of two angular momenta J = J1 + J2 , with j1=j2=1. Find the eigenstates of the total angular momentum I jm > in terms of the product states I j1 m1 j2 m2 > in two ways
(a) Make use of the tables of the Clebech _Gordan coefficients
(b) The state with m1 = m2 = 1 must be a state with j = m = 2 (why?). Apply J- repeatedly to this state to obtain all other states of j = 2. Form an appropriate linear combination of the two states with m1 + m2 =1 to obtain the state with j =1 , and m =1 . Find the other j = 1 states by applying J- repeatedly. Finally, find the j = m = 0 state by forming an appropriate linear combination of the three states with m1 + m2 = 0.
(c) Compare the results in (a) and (b).

I don't know how to use Clebech - Gordand coefficients, so please explain details using the table.

blue_leaf77
$$|j=2,m=1\rangle = \sqrt{\frac{1}{2}}|m_1=1\rangle |m_2=0\rangle + \sqrt{\frac{1}{2}}|m_1=0\rangle |m_2=1\rangle$$
$$|2,1\rangle = \sqrt{\frac{1}{2}}|1\rangle |0\rangle + \sqrt{\frac{1}{2}}|0\rangle |1\rangle$$