Why is j2 = 1/2 and not -1/2 in the coupled angular momenta equation?

[Dreak]

Hello, I have a small question about coupling of angular momenta.

When you have J² with J = J1 + J2; you change it to the form (dropping the hbar in all equations):

j² = j₁² + j₂² + j_1-j₂₊ + j₁₊j_2- + 2j_1zj_2z

now it's these j_1-j₂₊ I have a problem with.

Let's say you use it on |1/2;-1/2>. The answer is √[j1(j1+1) - m1(m1-1)] . √[j2(j2+1) - m2(m2 + 1)]

Ok, no problem. But apperently, the answer is: √[1/2(1/2+1) - 1/2(1/2-1)] . √ [ 1/2(1/2 + 1) - (-1/2)(-1/2 + 1)]

So the j2 = 1/2 and not -1/2 and I don't know why...
Unless it is because the angular momenta is quantisised from 0, 1, ...? Could that be it?
and what if you a further j- on the new basecomponent?

edit: nvm, found out :)

 

[DrClaude]

Let's say you use it on |1/2;-1/2>

That notation is a shorthand for $| m_1; m_2 \rangle$, where it is assumed that these are spin-1/2 particles, $j_1 = j_2 = 1/2$.