Superposition and product forms a) show that if ψ1(x, t) and ψ2(x, t) are solutions of the schrodinger equation 2marī + v(x, t) ) ψ(x, t) then the superposition ψ(x, t) ψ1(x, t) ψ2(x, t) is a solution too. b) consider now two schrödinger equations with different potential terms vm(x1, t) and ot 2m32,t) show that if ψ1(x1,t) is a solution of equation (2) and ψ2(t2,t) is a solution of equa- tion (3), then the product form ψ(x1, 22,t) = ψ1(m, t)ψ2(t2,t) is a solution of the schrödinger equation composed of the two equations above, namely c) replace in part b) all time derivatives θ/it by 02/9t2 and show that in this case the product form is not solution of the composite schrödinger equation. d) the composite schrödinger equation (4) in part b) describes a system of two inde- pendent particles. use the interpretation that x1,x2,t)12 is a probability density to show that the state of such a system must be in the product form ψ(x1,x2,t) ψ1 (x1, t) ψ2(x2,t). (note: this does disqualify the wave equation considered in part c) as a valid equation of motion for a system of independent particles.)