(a) Starting from , prove the triangle inequality for any two states of , naming each…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Hilbert space and Dirac notation

Problem

(a) Starting from ∥ϕ+ψ∥2=⟨ϕ+ψ∣ϕ+ψ⟩\|\phi+\psi\|^2=\langle\phi+\psi|\phi+\psi\rangle, prove the triangle inequality ∥ϕ+ψ∥≤∥ϕ∥+∥ψ∥\|\phi+\psi\|\le\|\phi\|+\|\psi\| for any two states of H\mathcal H, naming each property of the inner product you use. (b) Take ∣ϕ⟩=∣↑⟩|\phi\rangle=|{\uparrow}\rangle and ∣ψ⟩=35∣↑⟩+4i5∣↓⟩|\psi\rangle=\tfrac35|{\uparrow}\rangle+\tfrac{4i}{5}|{\downarrow}\rangle. Evaluate ∣⟨ϕ∣ψ⟩∣2|\langle\phi|\psi\rangle|^2 and ⟨ϕ∣ϕ⟩⟨ψ∣ψ⟩\langle\phi|\phi\rangle\langle\psi|\psi\rangle, then ∥ϕ+ψ∥\|\phi+\psi\| and ∥ϕ∥+∥ψ∥\|\phi\|+\|\psi\|. (c) Show that, for ∣ϕ⟩≠0|\phi\rangle\ne0, the triangle inequality is an equality exactly when ∣ψ⟩=λ∣ϕ⟩|\psi\rangle=\lambda|\phi\rangle with λ\lambda real and non-negative. The Schwarz inequality is an equality for ∣ψ⟩=i∣ϕ⟩|\psi\rangle=i|\phi\rangle; why is the triangle inequality not?

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