Quantum Mechanics: Concepts and Applications · Problem 2.14

Mathematical Tools of Quantum Mechanics — Nouredine Zettili

Physics — Quantum Mechanics · trace-of-operators

Problem

(a) Show that Tr(A^B^)=Tr(B^A^)\mathrm{Tr}(\hat{A}\hat{B}) = \mathrm{Tr}(\hat{B}\hat{A}). (b) Show that the trace of a commutator is always zero. (c) Illustrate the results shown in (a) and (b) on the following matrices: A=(8−2i4i0101−i−8i6i),B=(−i21−i61+i3i15+7i0).A = \begin{pmatrix} 8-2i & 4i & 0 \\ 1 & 0 & 1-i \\ -8 & i & 6i \end{pmatrix}, \quad B = \begin{pmatrix} -i & 2 & 1-i \\ 6 & 1+i & 3i \\ 1 & 5+7i & 0 \end{pmatrix}.

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