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Homework Help: Quantum State Tomography

  1. Dec 12, 2009 #1
    1. The problem statement, all variables and given/known data

    Assume there is a source of some pre-selected atoms. When measuring atoms of that source in a Stern-Gerlach, you find the following probabilities for a spin-up result:

    x-direction 5/6
    y-direction 5/6
    z-direction 1/3

    Which state would you ascribe to the source?

    2. Relevant equations

    ?

    3. The attempt at a solution

    Let the atoms be:

    [tex]\[ \left( \begin{array}{ccc}
    \alpha \\
    \beta \end{array} \right)\][/tex]

    Decomposing in the z-basis we get that the probability amplitudes for up and down are [tex]|\alpha|^2[/tex] and [tex]|\beta|^2[/tex], respectively.

    From this we find that [tex]\alpha = 1/\sqrt{3}[/tex] and [tex]\beta = \sqrt{2/3}[/tex]

    Ok this is where Im confused, my textbook says we need to do a measurement in all 3 directions, but I got these values with just the z result?

    If someone could explain this to me itd be a big help, im stuck :(
     
  2. jcsd
  3. Dec 12, 2009 #2

    Redbelly98

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    Staff Emeritus
    Science Advisor
    Homework Helper

    Be careful, you have not actually found α and β. You have found |α| and |β|. There is a complex phase factor still to be accounted for.
     
  4. Dec 13, 2009 #3
    Hmmm, ok well what I have here is, if |a|2 + |b|2 = 1, then |a|2 - |b|2 is a number between 1 and -1, so we write:

    |a|2 - |b|2 = cos 2x, where x is between 0 and pi/2

    then

    |a|2 = 1/2 (|a|2 + |b|2) + 1/2 (|a|2 - |b|2)
    |a|2 = 1/2 + 1/2 (cos 2x) = cos2x

    and similarly

    |b|2 = sin2x

    so then

    [tex]\alpha = e^i^\varphi cos x[/tex]
    [tex]\beta = e^i^\phi sinx[/tex]

    and then these are the phase factors you are talking about? I know they have magnitude 1, but Im not sure I know how to proceed from here :S
     
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