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Determing the composition of a state

  1. Jan 12, 2013 #1
    Hey,

    I have a question on determing the composition of a state of a system of composed of only two eigenvectors, the question is displayed below:

    determinev.png

    I initially assumed that the ket v was given by:

    [tex]|v>=a|\omega_{1}>+b|\omega_{2}>[/tex]

    Where 'a' and 'b' are constants which will determine the probability of either state. So we know the probability of the eigenvalues of ket v are given by the coefficients a and b in the equation:

    [tex]\mid <v|\hat{\Omega}|v>\mid^{2}[/tex]

    Where for the probability of attaining the eigenvalue ω(1) we have equation:

    [tex]\frac{1}{4}=a^{4}[/tex]

    Though I'm not sure this is correct, it implies

    [tex]a=\frac{1}{\sqrt{2}}\: ,\: b=\sqrt{\frac{\sqrt{3}}{2}}[/tex]

    I think I have made a mistake on my third equation...

    Thanks for any help,
    SK
     
  2. jcsd
  3. Jan 12, 2013 #2
    P(ω1) = |<ω1|v>|2 , so: a = 1/2 and b will be (3/4)1/2 multiplied by an arbitary phase factor.
     
  4. Jan 12, 2013 #3
    Right okay, this is the other way I done it but wasn't sure which way was correct... though this answer definitely makes more sense!... obviously.

    Thanks cosmic dust!
    SK
     
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