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Complex conjugation operator

  1. Mar 6, 2017 #1
    1. The problem statement, all variables and given/known data
    Hi, so I have been given the following operator in terms of 3 orthonormal states |Φi>

    A = |Φ2><Φ2| + |Φ3><Φ3| - i|Φ1><Φ2| - |Φ1><Φ3| + i|Φ2><Φ1| - |Φ3><Φ1|
    So I need to determine whether A is unitary and/or Hermitian and/or a projector and then calculate the eigenvalues and eigenfunctions in the |Φi> basis.

    The second question is to find eigenvalues and eigenfunctions of the complex conjugation operator acting on complex functions, Cα(x) = α*(x)

    2. Relevant equations


    3. The attempt at a solution
    So for the first one I said it is an operator because, it cannot be unitary since AAτ ≠ unit matrix and not hermitian since A ≠ A, but now I fail to show A2 = A in order to prove that it is actually a projector. please help if there is an easier way.

    The second part of the question am just failing to use that A in the formula A|Φi> = a|Φi> to find the eigenvalues and eigenfunctions.

    The second question I don't know where to even start.
    Please help, thank you very much.
     
  2. jcsd
  3. Mar 6, 2017 #2

    Orodruin

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    Please provide your actual attempt.

    What is the definition of an eigenfunction? How does complex conjugation act on a general complex function?
     
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