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Electron position at time t

  1. Sep 8, 2014 #1
    1. The problem statement, all variables and given/known data
    Consider the electron on a six site chain. The Hamiltonian is:

    ##
    H = \begin{pmatrix}
    0 & -1 & 0 & 0 & 0& 0\\
    -1 & 0 & -1 & 0 & 0& 0\\
    0 & -1 & 0 & -1 & 0& 0\\
    0 & 0 & -1 & 0 & -1& 0\\
    0 & 0 & 0 & -1 & 0& -1\\
    0 & 0 & 0 & 0 & -1& 0\\
    \end{pmatrix}##

    This is, I think, the computed value of H:
    ##H_{diagonal} = \begin{pmatrix}
    e^{-it} & 0 & 0 & 0 & 0& 0\\
    0 & e^{-i2t} & 0 & 0 & 0& 0\\
    0 & 0 & e^{-i2t} & 0 & 0& 0\\
    0 & 0 & 0 & e^{-i2t} & 0& 0\\
    0 & 0 & 0 & 0 & e^{-i2t}& 0\\
    0 & 0 & 0 & 0 & 0& e^{-it}\\
    \end{pmatrix}##

    If the electron starts in ##|x=0\rangle##, find where the electron is at time t.

    2. Relevant equations



    3. The attempt at a solution

    I'm just not sure how to go about this. What exactly does ##|x=0\rangle## mean. Do I need to use the Dirac notation and ##H|0\rangle##? Or is it that I need to multiply the diagonal matrix by

    ##\begin{pmatrix}
    1\\
    0 \\
    0 \\
    0 \\
    0 \\
    0 \\
    \end{pmatrix}##
     
  2. jcsd
  3. Sep 9, 2014 #2

    Simon Bridge

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    The electron starts at position x=0 in the 6 site chain.
    Other possible positions are, presumably, x=1, x=2,...x=5.
    You need to time-evolve that state.
     
  4. Sep 10, 2014 #3

    vela

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    That's not correct. I posted in your other thread.

    You gave a matrix representation for the Hamiltonian. What basis is this representation with respect to? Answering that might provide you the answer to your question about what ##|x=0\rangle## means.

     
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