22. (a) and (b) In the region 0 d2È 8Ä„2m
+ EÈ =0
dx2 h2
where E>0. If È2(x) =B sin2 kx, then È(x) =B sin kx, where B is another constant satisfying
B 2 = B. T hus d2È/dx2 = -k2B sin kx = -k2È(x) and
d2È 8Ä„2m 8Ä„2m
+ EÈ = -k2È + EÈ .
dx2 h2 h2
This is zero provided that
8Ä„2mE
k2 = .
h2
The quantity on the right-hand side is positive, so k is real and the proposed function satisfies
Schrödinger s equation. In this case, there exists no physical restriction as to the sign of k. It can
assume either positive or negative values. Thus
"
2Ä„
k = Ä… 2mE .
h