How to see that $[\textbf{p}^2,\textbf{L}^2]=[\textbf{p}^4,\textbf{L}^2]=0$ without doing any messy algebra?
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Suppose that AA is a vector operator, i.e its components transform as a vector under rotation,
[Li,Aj]=iϵijkAk.[Li,Aj]=iϵijkAk.
Using this definition gives [L,A2]=0[L,A2]=0, which just says that A2A2 is a rotational scalar. Then the commutator product rule implies [L2,A2]=[L2,A4]=0[L2,A2]=[L2,A4]=0 as desired.
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Suppose that AA is a vector operator, i.e its components transform as a vector under rotation,
[Li,Aj]=iϵijkAk.[Li,Aj]=iϵijkAk.
Using this definition gives [L,A2]=0[L,A2]=0, which just says that A2A2 is a rotational scalar. Then the commutator product rule implies [L2,A2]=[L2,A4]=0[L2,A2]=[L2,A4]=0 as desired.
hope it will help u if u like my ans. plzzzz mark me as brainileast
thnx dear
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