Expansion of an Eigenket

Question

What is the purpose of multiplying only one base ket by $e^{i\theta}$, when expanding an eigenket as a linear combination of its base kets?

Example: $|S_x; +\rangle = \frac{1}{\sqrt2}(|+\rangle + |-\rangle)$

Answer 1

Since you didn't ask about the $\sqrt {2} $ you seem to be okay with normalization. So really you should have a phase factor on both terms. So an $e^{i \beta} $ on the first one. But you can multiply everything by $e^{-i \beta} $ to make the phase factor only on the second term.

Edit: And then you see $e^{-i \beta} e^{i \theta} $ which you can rename as a new $e^{i \theta} $ for a replacement $\theta$