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Obtain Time evolution from Hamiltonian

  1. Jan 24, 2015 #1
    1. The problem statement, all variables and given/known data
    A quantum system with a ##C^3## state space and a orthonormal base ##\{|1\rangle, |2\rangle, |3\rangle\}## over which the Hamiltonian operator acts as follows:
    ##H|1\rangle = E_0|1\rangle+A|3\rangle##
    ##H|2\rangle = E_1|2\rangle##
    ##H|3\rangle = E_0|3\rangle+A|1\rangle##

    Build H and obtain the eigenvalues and the eigenvectors.

    If ##|\Phi((0)\rangle = |2\rangle## obtain ##|\Phi(t)\rangle##.

    If ##|\Phi((0)\rangle = |3\rangle## obtain ##|\Phi(t)\rangle##.
    2. Relevant equations


    3. The attempt at a solution
    I've managed to build the H matrix and obtain the eigenvectors.
    The second part is ##|\Phi(t)\rangle = e^{-\frac{i}{\hbar}E_1t}|2\rangle##
    It is the third part that I'm not sure. Should I build the change of basis matrix from the eigenvectors and apply it to the vector ##|3\rangle## expressed as ## |0 0 1\rangle##, so that I get its eigenvectors descomposition?
     
  2. jcsd
  3. Jan 24, 2015 #2

    mfb

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    2016 Award

    Staff: Mentor

    That should work, yes. Once you have the composition in eigenvectors, the time-evolution is easy to write down.
     
  4. Jan 25, 2015 #3
    Thanks mfb.

    Just a little side question: I've done the change of basis with ##|3\rangle' = S|3\rangle S^{-1}##. Could I have done ##|3\rangle' = S^\dagger|3\rangle S##?
     
    Last edited: Jan 25, 2015
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