Addition of Angular momentum

[Nuwbs]

Homework Statement

So I'm told I can't do it this way but I was wondering if anyone could clarify as to why? We're given $$|J=\frac{1}{2},M = \frac{1}{2}\!>$$ where $$j_1 = 1 \, and \, j_2 = \frac{1}{2}$$

Homework Equations

The Attempt at a Solution

So this can be composed as a linear combination:
$$| \frac{1}{2} \frac{1}{2}\!> = C_1 |1 1\!>|\frac{1}{2} -\frac{1}{2}\!> + C_2 |10\!> \frac{1}{2}\frac{1}{2}\!>$$
Applying the raising operator to both sides $$J_+$$ gives:
$$0 = C_1 |1 1\!>|\frac{1}{2} \frac{1}{2}\!> + \sqrt{2}C_2 |11\!> \frac{1}{2}\frac{1}{2}\!>$$ so that $$C_1 = -\sqrt{2}C_2 \, and \, C^2_1 + C^2_2 = 1 \, implies \, C_2 = \frac{1}{\sqrt3} \, and \, C_1 = \frac{\sqrt2}{\sqrt3}$$
But, I'm told this is wrong, why and thank you.

 

[vela]

Other than a sign mistake — one of the constants should be negative — it looks fine to me.

 

[Nuwbs]

I was told that this is true but that you cannot construct J=1/2 states directly, moreover that the coefficients are relative?