# Math Help - Two-Dimensional Hilbert Space = all pts. on Bloch Sphere

1. ## Two-Dimensional Hilbert Space = all pts. on Bloch Sphere

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, $H_2$) and the complex projective line. Now here is what I know:

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

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

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