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Interaction Hamiltonian

  1. Feb 5, 2010 #1
    1. The problem statement, all variables and given/known data

    Write out:

    [tex]H_{SE}(\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle)[/tex]

    and

    [tex]exp(-iH_{SE}t)(\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle)[/tex]


    Where:

    [tex]H_{SE}=\sum_{\alpha,j}\gamma(\alpha,j)P^{(\alpha)}\otimes\left|e_{j}\right\rangle\left\langle e_{j}\right|[/tex]

    and

    [tex]P^{(\alpha)}=\sum_{i_{\alpha}}\left|i_{\alpha}\right\rangle\left\langle i_{\alpha}\right|[/tex]


    ([tex]\left|i_{\alpha}\right\rangle[/tex] can be written [tex]\left|\right\alpha,i_{\alpha}\rangle[/tex] where alpha is a quantum number indexed by [tex]i_{\alpha}[/tex] )

    3. The attempt at a solution

    For the first part I'm fairly sure it comes out as:

    [tex]\sum_{\beta,j}\gamma(\beta,j)\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle[/tex]


    But the second part I am not sure of, is it something like:

    [tex](Cos(t)-i\gamma(\alpha,j)Sin(t))(\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle)[/tex]


    Thanks!
     
  2. jcsd
  3. Feb 5, 2010 #2
    In the first you should not summate over [itex]j[/itex] (and you need to explain why ;))

    For the second you first apply the Taylor expansion for the exponential. After that, compute:

    [itex]H_{SE}^2[/itex] followed by generalizing this to [itex]H_{SE}^n[/itex].
     
  4. Feb 5, 2010 #3
    Thanks for that.

    I'll have a bash at that.. although I honestly can't see why you wouldn't sum over j
     
  5. Feb 5, 2010 #4
    Oh wait... is it because the [tex]e_{j}[/tex] basis correspond to different alpha's but not i's?

    Edit: Actually on second thought that doesn't make sense because we are summing over alpha(beta).
     
  6. Feb 7, 2010 #5
    Can anyone else offer some more help?

    -I've been teaching myself dirac notation as part of my project this year. This is the first time I've looked at interaction Hamiltonians.
     
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