Principles of Quantum Mechanics · Problem 1.7.1

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · active-passive-transformations-trace

Problem

The trace of a matrix is defined to be the sum of its diagonal matrix elements Tr Ω=∑iΩii\text{Tr}\, \Omega = \sum_i \Omega_{ii} Show that (1) Tr(ΩΛ)=Tr(ΛΩ)\text{Tr}(\Omega\Lambda) = \text{Tr}(\Lambda\Omega) (2) Tr(ΩΛθ)=Tr(ΛθΩ)=Tr(θΩΛ)\text{Tr}(\Omega\Lambda\theta) = \text{Tr}(\Lambda\theta\Omega) = \text{Tr}(\theta\Omega\Lambda) (The permutations are cyclic). (3) The trace of an operator is unaffected by a unitary change of basis ∣i⟩→U∣i⟩|i\rangle \to U|i\rangle. [Equivalently, show Tr Ω=Tr(U†ΩU)\text{Tr}\,\Omega = \text{Tr}(U^\dagger\Omega U).]

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