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Poission Bracket of Angular Momentum

  • Thread starter cstalg
  • Start date
6
0
1. Homework Statement
Parts vii and viii of problem 2 of the attached file "hw5.pdf". It's too long and I'm too lazy to type it here.


2. Homework Equations
A useful equality of Levi-Civita symbol:

[itex]\epsilon_{ijk}\epsilon_{i'j'k}=\delta_{ii'}\delta_{jj'} - \delta_{ij'}\delta_{ji'}[/itex]


3. The Attempt at a Solution
I calculated the Poission brackets of those two expressions and none of them satisfies the definition of a rank two tensor given in the problem, although they are tensors according to the usual definition. The file "solutions5.pdf" contains the solution by the instructor or TAs. But I think the last equality of the proof of "fact 4" is incorrect: How can the following expression holds?

[itex]-\delta_{ji}f_k + \delta_{ki}f_j = \epsilon_{ijk}f_i[/itex]

If this is wrong, the solutions of parts vii and viii will be meaningless.

In quantum mechanics, scalars and vectors are defined using commutators [itex][L_i, V_j]=i\hbar\epsilon_{ijk}V_k[/itex], exactly the same algebra as in our problem. But no one defines Cartesian tensors using commutators/Poission brackets. I think Cartesian rank-two tensors could not be defined by simply generalizing the definition of a vector like in this problem.
 

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