a) The time-dependent Schr6dinger equation is of the form


a ∂Ψ/∂t =ĤΨ



Consider that a is an unspecified constant. Show that this equation has the following property.
Let H be the Hamiltonian of a system composed of two independent parts, so that
Ĥ(χ1,χ2)=Ĥ1(χ1)+Ĥ2(χ2)


and let the stationary states of system 1 be Ψ1(χ1,t) and those of system 2 be Ψ2(χ2,t). Then the
stationary states of the composite system are


Ψ(χ1,χ2) = Ψ1(χ1,t)Ψ2(χ2,t)

That is, show that this product form is a solution to the preceding equation for the given composite
Hamiltonian.
Such a system might be two beads that are invisible to each other and move on the same
straight wire. The coordinate of bead 1 is x1 and the coordinate of bead 2 is x2.
(b) Show that this property is not obeyed by a wave equation that is second order in time.
such as

a2 ∂2Ψ/∂t2 =ĤΨ

(c) Arguing from the Born postulate, show that the wavefunction for a system composed
of two independent components must be in the preceding product form, thereby disqualifying
. the wave equation in part (b) as a valid equation of motion for the wavefunctionΨ