(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let f(q, p), g(q, p) and h(q, p) be three functions in phase space. Let Lk =

ε_{lmk}q_{l}p_{m}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 [q_{j}, L_{k}], expressing your answer in terms of the permutation symbol.

(iv) Show [L_{j}, L_{k}] = q_{j}p_{k}−q_{k}p_{j }. Show also that the RHS satisfies q_{j}p_{k}−q_{k}pj =

ε_{ijk}L_{i}. Deduce [L_{i}, |L|^{2}] = 0.

[Hint: the identity ε_{ijk}ε_{klm}= δ_{il}δ_{jm}− δ_{im}δ_{jl}may be useful in (iv)]

2. Relevant equationsn/a

3. The 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 ε_{lmk}q_{l}

iv) my working is quite long, but I get [L_{j},L_{k}]=q_{j}p_{k}-q_{k}p_{j}=ε_{ijk}L_{i}as required.

The bit I'm having trouble with is the very last bit of the question, to deduce [L_{i}, |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:

[L_{i}, |L|^{2}]=[L_{i}, L_{j}L_{j}]=L_{j}[L_{i}, L_{j}]+[L_{i}, L_{j}]L_{j}=2L_{j}[L_{i}, L_{j}]

I'm not entirely sure where to go from here so any help (or pointing out of any glaring errors) would be great.

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# Poisson brackets and angular momentum

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