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Homework Help: Quantum Mechanics - quick question about probability

  1. Feb 26, 2012 #1
    Quantum - probability in a state
    I have an eigenvalue (d) and i need to find the probability of it in a state k.

    What is the equation?
    <k|d|k> ?

    I have spent some thought on this and it seems to simple.

    thanks
     
  2. jcsd
  3. Feb 26, 2012 #2

    Fredrik

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    Note that <k|d|k>=d<k|k>=d.

    I'm not sure what the exact question is. I'm guessing that what you have in mind is that the state preparation procedure has put the system in state |k>, that you're going to measure an observable D, and that d is an eigenvalue of D with a 1-dimensional eigenspace. Then the probability that the result of the D measurement will be d is ##|\langle d|k\rangle|^2##, where |d> is a normalized eigenstate of D with eigenvalue d.

    <k|D|k> is the average value you will get if you do this measurement on many identically prepared systems.
     
  4. Feb 26, 2012 #3
    If you have a state |k> then you want to take the inner product with the eigenbra <d|

    If you expand |k> in d eigens then
    |k> = <d|k>|d>
    (Sum over d)

    So <d|k> is the expansion coefficient for |d> or how much |d> is in |k> eg, it is the 'probability' of |k> as being in the eigenstate |d>

    Taking the inner product with your chosen <d'| gives

    <d'|k> = <d|k><d'|d>
    <d'|k> = <d'|k>
    (Sum over d)

    This happens since the eigenstates are orthogonal (I'm assuming they're orthogonal)
    The probability is then |<d'|k>|^2

    Do you understand what I have done here?
     
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