what are the angles , and ϕ values of the following quantum state?

Question

I need to find the coordinate and ϕ values of the quantum state on the bloch sphere $$ \left| \varphi \right>=\frac{1+i}{\sqrt{3}} \left| 0 \right> + {\sqrt{\frac{1}{3}}} \left| 1\right> $$

Answer 1

You have normalized state $$|\psi\rangle=\alpha|0\rangle + \beta|1\rangle$$

First, write the state as $$|\psi\rangle=\frac{\alpha}{|\alpha|}\left({|\alpha|}|0\rangle + \frac{\beta|\alpha|}{\alpha}|1\rangle\right)$$

The factor $$\frac{\alpha}{|\alpha|}$$ is a global phase and not important. Now you have $$\cos{\frac{\theta}{2}}=|\alpha|$$ which gives the value of $\theta$ and $$\sin{\frac{\theta}{2}}e^{i\phi}=\frac{\beta|\alpha|}{\alpha}$$ which gives the value of $\phi$