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Homework Help: QM two state system

  1. May 13, 2008 #1
    1. The problem statement, all variables and given/known data
    QM states of a system are described by linear super positions of two linearly independent state vectors psi1 and psi2. These two states are normalized but are NOT orthogonal to each other. A hermitian operator A actes on the two states in the following way.

    A|psi1> = 5|psi1>+3|psi2>
    A|psi2> = -3|psi1> - 5|psi2>

    Determine the eigenvalues and properly normalized eigenvectors of A.

    3. The attempt at a solution

    I attempted to introduce a general state |phi> = C1|psi1> + C2|psi2>, act A on |phi> and apply the normalization condition on |phi> to determine C1 and C2 but have gotten nowhere. I have found the value of <psi2|psi1>. Thanks for your help.
  2. jcsd
  3. May 13, 2008 #2


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    Science Advisor
    Homework Helper

    First find the eigenvalues and eigenvectors of A in the psi1, psi2 basis. The eigenvectors must be orthogonal since the operator is supposed to be hermitian. Now using that psi1 and psi2 are normalized, you should be able to normalize the eigenvectors.
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