Consider the following single-qubit state on the Block sphere

Question

$$\left| \varphi \right>=\frac{1+i}{\sqrt{3}} \left| 0 \right> + {\sqrt{\frac{1}{3}}} \left| 1\right> $$

I need to find the coordinate , and ϕ values of the quantum state.

Answer 1

To build on the answer already given by MKM, you should start of by showing that you can write your state as

$$\left|\varphi\right> = \sqrt{\frac{2}{3}} e^{i \frac{\pi}{4}} \left|0\right> + \sqrt{\frac{1}{3}} \left| 1 \right>,$$

from which the answer follows almost trivially. It might be helpful to remember that in Quantum Mechanics, a state is only defined up to an overall multiplicative phase. In other words, the states $\left|\psi\right>$ an $e^{i\delta}\left|\psi\right>$ are physically equivalent.

Answer 2

Given the fact, that any state on the Bloch sphere is given by (see Bloch sphere): $$\newcommand{\ket}[1]{|#1\rangle} \ket\psi=\cos(\theta/2)\ket 0+e^{i\phi}\sin(\theta/2)\ket 1$$ You just need to solve system of equations.