Problems finding the Eigenstates of a Hamiltonian I am solving the problem 5.6 from Zettili Quantum Mechanics book:
I have an Hamiltonian given by $\hat{H} = \epsilon \hat{\sigma} \cdot \hat{n}$ where $\hat{\sigma}$ represents the Pauli Matrices and $\hat{n}$ a arbitrary unit vector.
Using spherical coordinates I found the Eigenvalues, which are $E_{1} = \epsilon$ and $E_2 = -\epsilon$ but I can't find the Eigenvectors correctly. I find the following equation for the Eigenvectors equation:
$$ x \cos\theta + ye^{-i\phi} \sin\theta = E x$$
$$ x \sin{\theta}e^{i\phi} - y \cos\theta = E y$$
Where $E$ stands for the Eigenvalues in each case. For the first Eigenvalue I get the following relation between the x and y:
$$ \frac{x}{y} = \frac{e^{-i\phi}\sin\theta}{1-\cos\theta}$$ and from this point on I don't understand how the author proceeds with the solution. He writes that the previous relation is equal to this one: 
$$\frac{x}{y} =\frac{\cos\frac{\theta}{2} e^{-i\phi/2}}{\sin\frac{\theta}{2} e^{i\phi/2} }$$
And that the numerator is the first element of the eigenvalue and the denominator is the second element of the eigenvector. But I don't understand why.
 A: From your eigenvector equation we have:
$$\frac{x}{y} = e^{-i\phi}\frac{ \sin \theta}{1-\cos\theta}=e^{-i\phi}\frac{2\sin(\theta/2)\cos(\theta/2)}{2\cos^2(\theta/2)}=\frac{\sin(\theta/2)e^{-i\phi/2}}{\cos(\theta/2)e^{i\phi/2}}$$
Where I've used the double angle formulas. 
For your second question, keep in mind that for an eigenvalue-eigenvector pair $E,|v\rangle$, we have by definition:
$$\hat H|v \rangle = E|v \rangle$$
Multiplying both sides by some constant $\alpha \in \mathbb C$ gives:
$$\hat H(\alpha|v \rangle) = E(\alpha|v \rangle)$$
which means that $|v' \rangle = \alpha|v \rangle$ is also an eigenvector of $\hat H$ with the same eigenvalue. This means that you always have the freedom of multiplying the eigenvector by a constant when your calculating it, without worrying about it not being an eigenvector anymore. Another way you can look at this is that because of this freedom, any vector with components satisfying $x/y = {\sin(\theta/2)e^{-i\phi/2}}/{\cos(\theta/2)e^{i\phi/2}}$ is a valid eigenvector. 
In conclusion, the eigenvector with components $x={\sin(\theta/2)e^{-i\phi/2}}$ and $y={\cos(\theta/2)e^{i\phi/2}}$ is one valid eigenvector, since it gives the right $x/y$ ratio. Another (completely valid) choice would be something like $x={\sin(\theta/2)e^{-i\phi/2}}/{\cos(\theta/2)e^{i\phi/2}}$ and $y=1$. 
However, in Quantum Mechanics, people conventionally prefer to choose the normalized eigenstate (such that $\Vert v \Vert = 1$) between this infinite set of choices of eigenvectors; as this usually makes calculations (of probabilities, etc.) much simpler. Because of this, the author's choice is preferred by convention because $\Vert v \Vert^2 =|x|^2+|y|^2=\sin^2(\theta/2)+\cos^2(\theta/2)=1$.
