Quantum Mechanics: Concepts and Applications · Problem 2.4
Mathematical Tools of Quantum Mechanics — Nouredine Zettili
Physics — Quantum Mechanics · bra-ket-algebra
Problem
Consider the states ∣ψ⟩=3i∣ϕ1⟩−7i∣ϕ2⟩ and ∣χ⟩=−∣ϕ1⟩+2i∣ϕ2⟩, where ∣ϕ1⟩ and ∣ϕ2⟩ are orthonormal.
(a) Calculate ∣ψ+χ⟩ and ⟨ψ+χ∣.
(b) Calculate the scalar products ⟨ψ∣χ⟩ and ⟨χ∣ψ⟩. Are they equal?
(c) Show that the states ∣ψ⟩ and ∣χ⟩ satisfy the Schwarz inequality.
(d) Show that the states ∣ψ⟩ and ∣χ⟩ satisfy the triangle inequality.