A quantum particle moves freely in the positive x-direction following the equation ψ(x,t) = A[e^i(k₁^{x - ω}₁^t) + e^i(k₂^{x - ω}₂^t)]. Suppose k₂ = 4k₁ = 4k and a maximum of the probability distribution function |ψ(x,t)|² occurs at t = 0 and x = 0. i) Given that ω = ħk²/2m, determine the least positive value of x that gives a maximum in the probability distribution function at t = 4π/ω. ii) Use the result in part i) to determine the average speed in the negative x-direction of the probability distribution function. Also, determine the average speed using v_av = (ω₂ - ω₁)/(k₂ - k₁).