I'm considering a vector with complex coefficients.
$$\chi_0=c_1\begin{bmatrix}1 \\ 0\end{bmatrix}+c_2\begin{bmatrix}0 \\ 1\end{bmatrix} \,\,\,\,\,\,\,\,\,\, c_1,c_2 \in \mathbb{C}$$ I know that $|c_1|^2=\frac{2}{3}$ and that $|c_1|^2+|c_2|^2=1$. I would just like to know why I can reduce indetermination to just one parameter and not two. I would have written: $$\chi_0=\sqrt{\frac{2}{3}}e^{i\alpha}\begin{bmatrix}1 \\ 0\end{bmatrix}+\sqrt{\frac{1}{3}}e^{i\beta}\begin{bmatrix}0 \\ 1\end{bmatrix}\,\,\,\,\,\,\,\,\,\, \alpha,\beta \in \mathbb{R}$$ Instead I know that you can write too: $$\chi_0=\sqrt{\frac{2}{3}}\begin{bmatrix}1 \\ 0\end{bmatrix}+\sqrt{\frac{1}{3}}e^{i\gamma}\begin{bmatrix}0 \\ 1\end{bmatrix} \,\,\,\,\,\,\,\,\,\, \gamma \in \mathbb{R}$$ but I can not justify this passage mathematically.
There is no justification from a mathematical point of view. It's all quantum mechanics. The expectation value of any hermitian operator does not change if you change the phase of the wavefunction. So from the point of view of quantum mechanics $\chi_0$, and $\chi_oe^{i\alpha}$, with $\alpha\in\mathbb R$, are the same wavefunction.