Discussion
The commutator of the orbital angular momentum operators [L+, L-], where L± = Lx ± iLy is :
Answer: D
\([L_{+}, L_{-}] = 2\hbar L_{z}\).
Expand using \([L_{x}, L_{y}] = i\hbar L_{z}\):
\[ \begin{aligned} [L_{x} + iL_{y}, L_{x} - iL_{y}] &= -i[L_{x}, L_{y}] + i[L_{y}, L_{x}] \\ &= -2i[L_{x}, L_{y}] \\ &= -2i(i\hbar L_{z}) = 2\hbar L_{z} \end{aligned} \]
The commutator is not zero because \(L_{x}\) and \(L_{y}\) do not commute, and the cyclic algebra produces \(L_{z}\), not \(L_{x}\) or \(L_{y}\).
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