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Hermitian operator matrix

  1. Aug 30, 2016 #1
    1. The problem statement, all variables and given/known data
    Eigenvalues of the Hamiltonian with corresponding energies are:
    Iv1>=(I1>+I2>+I3>)/31/2 E1=α + 2β
    Iv2>=(I1>-I3>) /21/2 E2=α-β
    Iv3>= (2I2> - I1> I3>)/61/2 E3=α-β

    Write the matrix of the Hamiltonian in the basis of the orthonormalized vectors I1>, I2>, I3>

    If in t=0, system is in the state I1>, what is the wave function in t?

    2. Relevant equations
    Hij = <ilHlj>

    3. The attempt at a solution
    Although I know that energy is the eigenvalue of the Hermitian operator, I am not sure how to incorporate that in this certain problem. I have used mentioned equation for previous problems, but I always had the form of the operator. With only eigenvectors and eigenvalues I am stuck and don't even know how to begin solving this.
    Last edited: Aug 30, 2016
  2. jcsd
  3. Aug 30, 2016 #2


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    Staff: Mentor

    There seems to be a part of the question that is missing. Can you write it out fully?
  4. Aug 30, 2016 #3
    I appologize. I have written it now
  5. Sep 1, 2016 #4


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    Staff: Mentor

    One way to go about this is to start by writing the Hamiltonian in the |v1>, |v2>, |v3> basis, then applying the proper transformation operation to "rotate" the Hamiltonian to the |1>, |2>, |3> basis.
  6. Sep 2, 2016 #5
    But unfortunately I do not know how to do that
  7. Sep 8, 2016 #6


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    Science Advisor
    Homework Helper
    Gold Member

    The straightforward way to do it is
    1. Find |1>, |2> and |3> as linear combinations of |v1>, |v2> and |v3> and verify that they are orthonormal.
    2. Note that H|v1> = (α + 2β) |1> and get similar expressions for H operating on the other two v's.
    3. Calculate things like < 1 | H | 2 > using the linear combinations from item 1 and substituting from item 2.
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