cstalg
Mar23-08, 12:08 PM
1. The problem statement, all variables and given/known data
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. Relevant equations
A useful equality of Levi-Civita symbol:
\epsilon_{ijk}\epsilon_{i'j'k}=\delta_{ii'}\delta_ {jj'} - \delta_{ij'}\delta_{ji'}
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?
-\delta_{ji}f_k + \delta_{ki}f_j = \epsilon_{ijk}f_i
If this is wrong, the solutions of parts vii and viii will be meaningless.
In quantum mechanics, scalars and vectors are defined using commutators [L_i, V_j]=i\hbar\epsilon_{ijk}V_k, 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.
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. Relevant equations
A useful equality of Levi-Civita symbol:
\epsilon_{ijk}\epsilon_{i'j'k}=\delta_{ii'}\delta_ {jj'} - \delta_{ij'}\delta_{ji'}
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?
-\delta_{ji}f_k + \delta_{ki}f_j = \epsilon_{ijk}f_i
If this is wrong, the solutions of parts vii and viii will be meaningless.
In quantum mechanics, scalars and vectors are defined using commutators [L_i, V_j]=i\hbar\epsilon_{ijk}V_k, 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.