Quantum Mechanics: Concepts and Applications · Problem 2.15

Mathematical Tools of Quantum Mechanics — Nouredine Zettili

Physics — Quantum Mechanics · matrix-operations-kets-bras-operators

Problem

Consider a matrix AA (which represents an operator A^\hat{A}), a ket ∣ψ⟩|\psi\rangle, and a bra ⟨ϕ∣\langle\phi|: A=(53+2i3i−i3i81−i14),∣ψ⟩=(−1+i32+3i),⟨ϕ∣=(6−i5).A = \begin{pmatrix} 5 & 3+2i & 3i \\ -i & 3i & 8 \\ 1-i & 1 & 4 \end{pmatrix}, \quad |\psi\rangle = \begin{pmatrix} -1+i \\ 3 \\ 2+3i \end{pmatrix}, \quad \langle\phi| = \begin{pmatrix} 6 & -i & 5 \end{pmatrix}. (a) Calculate the quantities A∣ψ⟩A|\psi\rangle, ⟨ϕ∣A\langle\phi|A, ⟨ϕ∣A∣ψ⟩\langle\phi|A|\psi\rangle, and ∣ψ⟩⟨ϕ∣|\psi\rangle\langle\phi|. (b) Find the complex conjugate, the transpose, and the Hermitian conjugate of AA, ∣ψ⟩|\psi\rangle, and ⟨ϕ∣\langle\phi|. (c) Calculate ⟨ϕ∣ψ⟩\langle\phi|\psi\rangle and ⟨ψ∣ϕ⟩\langle\psi|\phi\rangle; are they equal? Comment on the differences between the complex conjugate, Hermitian conjugate, and transpose of kets and bras.

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