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Help with Poisson Brackets

  1. Nov 29, 2014 #1
    1. The problem statement, all variables and given/known data
    Consider the motion of a particle with charge e in a homogenous magnetic field B_i. The Hamiltonian for this problem is $$H = \frac{1}{2m} \sum_{i=1}^{i=3} \left[ p_i - \frac{e}{2}\epsilon _{ijk}B_j x_k\right]^2.$$ By calculating the Poisson brackets, show that the transformation $$p_i \rightarrow p_i - \frac{e}{2}\epsilon _{ijk}B_j x_k$$ is not canonical.

    2. Relevant Equations
    None that aren't already given.

    3. Attempt at Solution
    I feel decently comfortable using Poisson brackets, however I am not sure how to perform this calculation because of the summation from i=1 to i=3. My attempt was to add each component, so start with x_i and p_i and perform the poisson bracket and get 1, then repeat for x_j and p_j, and for x_k and p_k. Since each yields one, I got a total of 3. To be canonical it must equal 1. However, I really don't think this method is correct. Thanks for any help!
     
  2. jcsd
  3. Dec 4, 2014 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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