How can I write a normalized spin superposition in a simplified form?

[Ananthan9470]

If I have an normalized spin superposition |ψ> = (a₁+a₂i) |1> + (b₁+b₂i) |2>, and asked to write it in the form |ψ> = cosθ|1> + e^iΦsinθ|2>, how do I proceed?
My main problem is that no matter what I try, I can't seem to get rid of some complex component that shows up in the coefficient of |1>. I can try to get rid of that after taking the common phase factor from both the vecotrs but after I take it out, am I allowed to just ignore it?

 

[blue_leaf77]

I can try to get rid of that after taking the common phase factor from both the vecotrs but after I take it out, am I allowed to just ignore it?

If the question does not say anything about that, I would factor out the common phase factor to make the coefficient of $|1\rangle$ real. No matter what you try, the two expressions given in the question cannot be equated.

 

[Strilanc]

You're probably allowed to adjust the superposition's global phase (since the global phase is an unobservable artifact of the representation).

Try multiplying by the expression by the conjugate of $| 1 \rangle$'s coefficient, i.e. by $a_1 - a_2 i$. That's how you normally cancel out the phase of a complex coefficient. Then it's just a matter of renormalizing the length.