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(Quantum mec.) Probability of measuring a given value of an observable

  1. Feb 2, 2014 #1
    1. The problem statement, all variables and given/known data

    Given the following hamiltonian and the observable [itex]\widehat{B}[/itex]
    4zsXGft.png

    find the possible energy levels ([itex]a[/itex] is a real constant). If the state is in it's fundamental state what's the probability of measuring [itex]b_{1}[/itex], [itex]b_{2}[/itex] and [itex]b_{3}[/itex]?


    2. Relevant equations



    3. The attempt at a solution

    To find the energy levels I simply calculated the eigenvalues of the matrix. I got:

    [itex]E_{1}=E_{0}-\sqrt{2}a[/itex]
    [itex]E_{3}=E_{0}[/itex]
    [itex]E_{2}=E_{0}+\sqrt{2}a[/itex]

    Next I found the eigenvector associated with the eigenvalue [itex]E_{1}[/itex] to find the fundamental state. I got:
    [itex]v_{1}=\frac{1}{2}(1,\sqrt{2},1)[/itex]

    I don't know how to solve it from here tho.. Am I doing something wrong?
    The solutions are 1/4, 1/2 and 1/4, respectively.

    Thanks.
     
  2. jcsd
  3. Feb 2, 2014 #2

    TSny

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    Homework Helper
    Gold Member

    Hi Jalo. All looks good. You are almost there. (If the answer were a snake it would bite you. :smile:)

    Note that you can write ##v_1 = \frac{1}{2}(1, \sqrt{2}, 1)## as ##v_1 = \frac{1}{2}(1, 0, 0) + \frac{1}{\sqrt{2}}(0, 1, 0) + \frac{1}{2}(0, 0, 1)##.

    What do the individual states ##(1, 0, 0), (0, 1, 0),## and ##(0, 0, 1)## represent? Note that you are using a basis where B is diagonal.
     
  4. Feb 4, 2014 #3
    Thank you very much! I was so close, should have figured it out! :P
     
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