Superposition of states, spin 1/2

Homework Statement

Two particles, their spin are 1/2.

The hamiltonian is ##H=\gamma s_1 \cdot s_2##

At t=0, the state ##|\alpha(0)>## is such as ##s_{1z}|\alpha(0)>=\hbar/2 |\alpha(0)>## and ##s_{2z}|\alpha(0)>=\hbar/2 |\alpha(0)>##. Find the state ##|\alpha(0)>##.

2. The attempt at a solution
I think that ##|\alpha(0)>=|1,1>## in the base of total angular momentum, i.e. |1/2, 1/2> in the other one. But the correct result seems to be ##|\alpha(0)>=1/ \sqrt2 [|1,1; 1/2, 1/2> + 1/ \sqrt 2 (|1,0; 1/2, 1/2>+|0,0; 1/2, 1/2>)]##.. what's wrong?

I used the following table:

blue_leaf77
Homework Helper
##|\alpha(0)>=1/ \sqrt2 [|1,1; 1/2, 1/2> + 1/ \sqrt 2 (|1,0; 1/2, 1/2>+|0,0; 1/2, 1/2>)]##.
I don't see why an unnormalized state should be the correct answer, besides which bases does it use on the RHS?

EDIT: No, I didn't pay a closer attention on the parentheses, yes this state is normalized.

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I haven't undestood the result in the book. Can you explain me how you would solve this point?

I haven't understood why in the result appear the states |1,0;1/2,1/2> and |0,0;1/2,1/2>. They don't have s1z=1/2 AND s2z=1/2. They are linear combination of ##s_{1z}=\pm 1/2## AND ##s_{2z}=\pm 1/2##!!! :(

blue_leaf77