- #1
Rhi
- 10
- 0
Homework Statement
Let f(q, p), g(q, p) and h(q, p) be three functions in phase space. Let Lk =
εlmkqlpm be the kth component of the angular momentum.
(i) Define the Poisson bracket [f, g].
(ii) Show [fg, h] = f[g, h] + [f, h]g.
(iii) Find [qj , Lk], expressing your answer in terms of the permutation symbol.
(iv) Show [Lj , Lk] = qjpk−qkpj . Show also that the RHS satisfies qjpk−qkpj =
εijkLi. Deduce [Li, |L|2] = 0.
[Hint: the identity εijkεklm = δilδjm − δimδjl may be useful in (iv)]
Homework Equations
n/aThe Attempt at a Solution
i) [f,g]=[itex]\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}[/itex]
ii) easy to show from the definition in i)
iii) after a bit of working, I get εlmkql
iv) my working is quite long, but I get [Lj,Lk]=qjpk-qkpj=εijkLi as required.
The bit I'm having trouble with is the very last bit of the question, to deduce [Li, |L|2] = 0.
Since it's only a small part of the question, it seems as though this part should be fairly simple so maybe I'm overlooking something, but I don't get 0. This is my working:
[Li, |L|2]=[Li, LjLj]=Lj[Li, Lj]+[Li, Lj]Lj=2Lj[Li, Lj]
I'm not entirely sure where to go from here so any help (or pointing out of any glaring errors) would be great.