Using the Born approximation to determine the one-dimensional reflection coefficient R for a potential energy V(x) that vanishes everywhere except in the vicinity of the origin:
(a) Show that we can write the solution to the one-dimensional Schrödinger equation in the form
ψ(x)=Aeikx+∫dx′G(x,x′)ℏ22mV(x′)ψ(x′)
where
∂x2∂2G(x,x′)+k2G(x,x′)=δ(x−x′)
(b) Since G satisfies a second-order differential equation, G must be a continuous function and, in particular, it must be continuous at x=x′. By integrating the differential equation for G from just below to just above x=x′, show that the first derivative of G is discontinuous at x=x′ and that it satisfies
(∂x∂G)x=x+′−(∂x∂G)x=x−′=1
Then show that one solution for G is given by
G=⎩⎨⎧2ik1eik(x−x′)2ik1e−ik(x−x′)x>x′x<x′
(c) Substitute this expression for G into the equation for ψ in (a). Show that in the Born approximation
ψx→−∞Aeikx+Ae−ikx∫−∞∞dx′2ike2ikx′ℏ22mV(x′)
and that consequently
R=ikℏ2m∫−∞∞dx′e2ikx′V(x′)2
(d) For the potential barrier
V(x)={V000<x<aelsewhere
the exact reflection coefficient is given by R=1−T with
T=1+[V02/4E(E−V0)]sin2(2m/ℏ2)(E−V0)a1
Show that the exact result for R in the limit V0/E≪1 agrees with the result of the Born approximation.