Question

5. Consider a wavefunction which is a linear superposition of 2 plane waves $(x, t) ΑΨ1(x, t) + BV2(x, t) where ,,-exp [i(kix-wit)] and with A and B real constants (a) Show that (x, t) is a solution of the time-dependent Free Particle Schrödinger Equa- tion. Note: This requires establishing relationships between parameters of V (b) Show that What is it's maximum? It's minimum?

Answer 1

psi (x, t) = A psi_1 (x, t) + B psi_2 (x, t) with psi_1 = exp [i (k_1 x - w_1 t)] psi_2 = exp [i (k_2 x - w_2 t)] Time dependent eqn for free particle: i hwithstroke partial differential/partial differential t psi = p^2/2m d^2 psi/dx^2 = -hwithstroke_2/2m d^2/dx^2 psi so, partial differential psi/partial differential t = A (- i w_1) psi_1 + B (- i w_2) psi_2 d psi/dx = A (i k_1) psi_1 + B (ik_2) psi_2 d^2 psi/dx^2 = A (ik_1)^2 psi_1 + B (ik_2)^2 psi_2 = -Ak^2_1 psi_1 - B k^2_2 psi_2 so, i hwithstroke (- i) [A w_1 psi_1 + B w_2 psi_2] = hwithstroke^2/2m [A k^2_1 psi_1 + B k^2_2 psi_2] Equating the coefficients of A & B separately. hwithstroke w_1 = hwithstroke^2 k^2_1/2m & hwithstroke w_2 = hwithstroke^2 k^2_2/2m so, psi (x, t) is a solution of time dependent eqn provided the above two equations satisfied. psi* (x, t) = A r^- i (k_1 x - w_1 t)