Results 1 to 1 of 1

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

  1. #1
    Newbie
    Joined
    Oct 2012
    From
    District of Columbia
    Posts
    16

    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?
    Last edited by ncshields; February 6th 2014 at 01:20 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Stochastic PDE, infinite dimensional Hilbert Space [HARD]
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: May 10th 2011, 08:54 AM
  2. Proof of Hilbert space being finite-dimensional
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 11th 2010, 12:51 AM
  3. Inverse of Mapping from Hilbert Space to Hilbert Space exists
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 2nd 2009, 09:15 PM
  4. unit sphere in an infinite dimensional Banach space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 3rd 2009, 01:15 AM
  5. Infinite dimensional Hilbert spaces
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: February 10th 2006, 03:35 PM

Search Tags


/mathhelpforum @mathhelpforum