Quantum Mechanics: Concepts and Applications · Problem 2.5

Mathematical Tools of Quantum Mechanics — Nouredine Zettili

Physics — Quantum Mechanics · orthogonality-condition

Problem

Consider two states ∣ψ1⟩=2i∣ϕ1⟩+∣ϕ2⟩−a∣ϕ3⟩+4∣ϕ4⟩|\psi_1\rangle = 2i|\phi_1\rangle + |\phi_2\rangle - a|\phi_3\rangle + 4|\phi_4\rangle and ∣ψ2⟩=3∣ϕ1⟩−i∣ϕ2⟩+5∣ϕ3⟩−∣ϕ4⟩|\psi_2\rangle = 3|\phi_1\rangle - i|\phi_2\rangle + 5|\phi_3\rangle - |\phi_4\rangle, where ∣ϕ1⟩|\phi_1\rangle, ∣ϕ2⟩|\phi_2\rangle, ∣ϕ3⟩|\phi_3\rangle, and ∣ϕ4⟩|\phi_4\rangle are orthonormal kets, and where aa is a constant. Find the value of aa so that ∣ψ1⟩|\psi_1\rangle and ∣ψ2⟩|\psi_2\rangle are orthogonal.

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