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## 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 [tex] |J=\frac{1}{2},M = \frac{1}{2}\!> [/tex] where [tex] j_1 = 1 \, and \, j_2 = \frac{1}{2} [/tex]

## Homework Equations

## The Attempt at a Solution

So this can be composed as a linear combination:

[tex] | \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}\!> [/tex]

Applying the raising operator to both sides [tex] J_+ [/tex] gives:

[tex] 0 = C_1 |1 1\!>|\frac{1}{2} \frac{1}{2}\!> + \sqrt{2}C_2 |11\!> \frac{1}{2}\frac{1}{2}\!> [/tex] so that [tex] 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} [/tex]

But, I'm told this is wrong, why and thank you.