Introduction to Quantum Mechanics · Problem A.17

Afterword — David J. Griffiths

Physics — Quantum Mechanics · matrices

Problem

Prove that Tr(T1T2)=Tr(T2T1)\text{Tr}(\mathsf{T}_1 \mathsf{T}_2) = \text{Tr}(\mathsf{T}_2 \mathsf{T}_1). It follows immediately that Tr(T1T2T3)=Tr(T2T3T1)\text{Tr}(\mathsf{T}_1 \mathsf{T}_2 \mathsf{T}_3) = \text{Tr}(\mathsf{T}_2 \mathsf{T}_3 \mathsf{T}_1), but is it the case that Tr(T1T2T3)=Tr(T2T1T3)\text{Tr}(\mathsf{T}_1 \mathsf{T}_2 \mathsf{T}_3) = \text{Tr}(\mathsf{T}_2 \mathsf{T}_1 \mathsf{T}_3), in general? Prove it, or disprove it. Hint: The best disproof is always a counterexample---the simpler the better!

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