Ok, I have read a lot of material online to suggest that in Quantum Mechanics the Bloch Sphere = the Riemann Sphere = complex projective line.

I need to try to prove a bijective relationship between a two-dimensional Hilbert Space (i.e, $\displaystyle H_2$) and the complex projective line. Now here is what I know:

A projective hilbert space is noted: $\displaystyle P(H_n) = \mathbb{C}P^{n-1}$. So for $\displaystyle H_2$ we have $\displaystyle P(H_2) = \mathbb{C}P^1$. This is the complex projective line.

The way I understand it, an element of $\displaystyle \mathbb{C}P^1$ is $\displaystyle {\lambda(\alpha|0\rangle + \beta|1\rangle)|\lambda \in \mathbb{C}}$, and an element of $\displaystyle H_2$ is $\displaystyle \alpha|0\rangle + \beta|1\rangle$.

How do I go about showing 1-1 and onto?