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[Lx,H] = [yPz,V] - [ZPy,V] and by expanding using and the relation [AB,C] = A[B,C] + [A,C]B and the Pz = hbar/i * d/dz, etc. get

[Lx,H] = hbar/i * ( yf dV/dz - zf dV/y) where f is a test function.

I have not used the fact that V is a function of r only and I can't take this any further.

I know that ( yf dV/dz - zf dV/y) must equal zero but have no idea how to prove it. Please help with a suggestion.

(I believe that H commutes with each component Lx, Ly and Lz and not just L so I don't think the terms above will cancel with terms from [Ly,H] and [Lz,H].)