Show that the set of functions k (x) = e ikx (for real k) are the

Show that the set of functions ψ_k(x) = e^ikx (for real k) are the solutions to the condition ψ(x+δ) = e^iθ(δ)ψ(x) (for real θ). use sequential translations by δ₁ and δ₂ to show that θ must depend linearly on δ, then expand this condition about δ = 0. Show that if k were complex, then the probability dP/dx = |ψk(x)|² would grow without bound as x approaches either +∞ or −∞, corresponding to unphysical boundary conditions.