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Homework Help: Eigenvectors of a 2D hermitian operator (general form)

  1. Feb 7, 2012 #1
    1. The problem statement, all variables and given/known data
    Calculate the eigenvectors and eigenvalues of the two-dimensional
    matrix representation of the Hermitean operator [itex]\hat{O}[/itex]
    given by

    |v_k'>[itex]\left(O[/itex]|v_k>= {{O_11,O_12},{O_21,O_22}}


    where all Oij are real. What does Hermiticity imply for the o -
    diagonal elements O12 and O21? For the eigenvectors you may
    assume that the two base vectors read

    v_1= (1) v_2= (0)
    ........(0)...........(1)

    2. Relevant equations
    None


    3. The attempt at a solution

    So, I know how to work out eigenvectors/values:

    |A-λI|=0

    so, I end up with (O11-λ)(O22-λ)-O12O21=0 as the characteristic equation

    with solution λ= 1/2(O22+O11)±1/2√((O22+O11)^2-4(O11O22-O21O12))


    the matrix O, being hermitian, makes O12=O21*

    but how do I progress on to an actual answer?
     
  2. jcsd
  3. Feb 7, 2012 #2

    vela

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    Since the matrix is real, you know that O21* = O21. You should be able to show that
    $$\lambda = \frac{(O_{11}+O_{22}) \pm \sqrt{(O_{11}-O_{22})^2 + 4O_{12}^2}}{2}$$ That's about as simple as you can get it. Now you want to solve for the eigenvectors. You just have to grind through the algebra.
     
  4. Feb 7, 2012 #3
    But it just ends up as a horrible mess which cannot be the answer!
     
  5. Feb 8, 2012 #4

    vela

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    It's not that bad. You need to adjust your threshold for pain. :wink:
     
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