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Matrix representation of certain Operator

  1. Aug 21, 2016 #1
    1. The problem statement, all variables and given/known data
    Vectors I1> and I2> create the orthonormal basis. Operator O is:
    O=a(I1><1I-I2><2I+iI1><2I-iI2><1I), where a is a real number.
    Find the matrix representation of the operator in the basis I1>,I2>. Find eigenvalues and eigenvectors of the operator. Check if the eigenvectors are orthonormal.


    2. Relevant equations

    Av=λv

    3. The attempt at a solution

    My problem is concerning the first part of this excercise. I'm not really familiar with this notation of the operator and not sure how I should get the matrix. I have tried improvisation and got the matrix

    2a^2 -2a^2
    -2a^2 2a^2

    When I tried to calculate eigenvalues, I didn't get anything reasonable, so I believe that my matrix is wrong. Please help me regarding this problem, once I have the right matrix I will not have the problem finding eigenvalues nor eigenvectors.
     
  2. jcsd
  3. Aug 21, 2016 #2

    blue_leaf77

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    Hint: The matrix element ##O_{ij}## of an operator ##O## is given by ##\langle i|O|j\rangle##.
    That is actually the vector form of the relation for a matrix ##M##
    $$
    M = \sum_i\sum_j M_{ij} c_i r_j
    $$
    where ##c_i## is a column matrix containing 1 as the i-th element and zero otherwise and ##r_j## is a row matrix containing 1 as the j-th element and zero otherwise.
     
  4. Aug 21, 2016 #3
    Thank you very much for your reply. I know the formula for the matrix element but have problem working it out with this notation. I was trying to find examples which include this notation, but without any luck.
     
  5. Aug 21, 2016 #4

    blue_leaf77

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    What's the problem, for example ##\langle 1 |O| 2 \rangle = i a\langle 1| 1 \rangle \langle 2 |2 \rangle = ia##.
     
    Last edited: Aug 21, 2016
  6. Aug 21, 2016 #5
    I don't understand how you got i<1l1><2l2> and also, what to do with that scalar a in front of the bracket
     
  7. Aug 21, 2016 #6

    blue_leaf77

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    From ##\langle 1 |O| 2 \rangle##, replace ##O## with the form you are given with in the first post and then make use of the fact that ##|1\rangle## and ##|2\rangle## are orthonormal.
    Sorry I forgot to add ##a##. Corrected.
     
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