Help with Poisson Brackets

[mc0210]

Homework Statement

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.

Homework 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!

 

[mark726]

Thank you for your question. I am happy to help you with this problem.

Firstly, let's review the definition of a canonical transformation. A canonical transformation is a transformation of coordinates and momenta that preserves the form of Hamilton's equations. In other words, the Poisson bracket of the new coordinates and momenta should equal 1, as you correctly stated.

Now, let's look at the transformation given in the problem: $$p_i \rightarrow p_i - \frac{e}{2}\epsilon _{ijk}B_j x_k$$
We can rewrite this transformation as: $$p_i \rightarrow p_i + \frac{e}{2}\epsilon _{ijk}x_kB_j$$
Note that this transformation involves both the coordinates and momenta, which means we need to consider both in our calculation of the Poisson bracket.

As you correctly stated, we need to calculate the Poisson bracket for each component (i,j,k) separately. Let's start with i = 1:
$$\{x_1, p_1 + \frac{e}{2}\epsilon _{ijk}x_kB_j\} = \frac{\partial x_1}{\partial x_1}\frac{\partial p_1}{\partial p_1} + \frac{\partial x_1}{\partial p_1}\frac{\partial p_1}{\partial x_1} + \frac{\partial x_1}{\partial x_2}\frac{\partial p_1}{\partial p_2} + \frac{\partial x_1}{\partial p_2}\frac{\partial p_1}{\partial x_2} + \frac{\partial x_1}{\partial x_3}\frac{\partial p_1}{\partial p_3} + \frac{\partial x_1}{\partial p_3}\frac{\partial p_1}{\partial x_3}$$
Note that we only need to consider the terms that involve p_1 and x_1, as all other terms will evaluate to 0 due to the Kronecker delta function. Also, since the coordinates and momenta are independent, we can set the partial derivatives equal to 0. This gives us:
$$\{x_1, p_1 + \frac{e}{2}\epsilon _{ijk}x_kB_j