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Canonical Transformation/Poisson Brackets

  1. Jun 6, 2009 #1
    1. The problem statement, all variables and given/known data
    I am trying to show that [tex] [q_j, p_k] = \delta_{jk} [/tex] (this is part of exercise 2.7.3 from Shankar's QM). I'm having difficulties with the summation notation.

    2. Relevant equations



    3. The attempt at a solution
    [tex] [q_j, p_k] = \sum_{k} (\frac{\partial q_j}{\partial q_k} \frac{\partial p_k}{\partial p_k} - \frac{\partial q_j}{\partial p_k} \frac{\partial p_j}{\partial q_k}[/tex] [tex] = \sum_{k} - \delta_{jk} = \delta_{jk} ??[/tex]
    I'm not so confident on my choice of 'k' as the summation variable. It seems to me the summation should not disappear like that. If I am interpreting this correctly, the negative sign isn't such a big deal... Can anyone check my work, I don't think I am doing it correctly
     
    Last edited: Jun 6, 2009
  2. jcsd
  3. Jun 6, 2009 #2
    Yes, you have to be careful about that. Left, the [tex]p_k[/tex] carries an index k. That means you shouldn't use k as a summation dummy index on the right hand side. Try working it out starting from the following:

    [tex] [q_j,p_k] = \sum_n \left( \frac{\partial q_j}{\partial q_n}\frac{\partial p_k}{\partial p_n} - \frac{\partial q_j}{\partial p_n}\frac{\partial p_k}{\partial q_n}\right)[/tex]

    EDIT: mixed up the indices myself...
     
    Last edited: Jun 6, 2009
  4. Jun 6, 2009 #3

    Cyosis

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    Homework Helper

    There is more amiss here. The indices of the numerators are wrong as well. (we're talking about Poisson brackets here right?)

    [tex]
    [q_j,p_k]=\sum_n \left( \frac{\partial q_j}{\partial q_n}\frac{\partial p_k}{\partial p_n} - \frac{\partial q_j}{\partial p_n}\frac{\partial p_k}{\partial q_n}\right)
    [/tex]
     
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