21. (a) and (b) Schrödinger s equation for the region x>Lis
d2È 8Ä„2m
+ [E - U0] È =0 ,
dx2 h2
where E -U0 < 0. If È2(x) =Ce-2kx, then È(x) =C e-kx, where C is another constant satisfying
C 2 = C. Thus d2È/dx2 =4k2C e-kx =4k2È and
d2È 8Ä„2m 8Ä„2m
+ [E - U0] È = k2È + [E - U0] È .
dx2 h2 h2
This is zero provided that
8Ä„2m
k2 = [U0 - E] .
h2
The quantity on the right-hand side is positive, so k is real and the proposed function satisfies
Schrödinger s equation. If k is negative, however, the proposed function would be physically unre-
alistic. It would increase exponentially with x. Since the integral of the probability density over the
entire x axis must be finite, È diverging as x "would be unacceptable. Therefore, we choose

2Ä„
k = 2m (U0 - E) > 0 .
h