3.3 Solutions 1933.44 (a)
Fig. 3.18Penetration of a
rectangular barrier
(b) Region 1,x< 0
d^2 ψ
dx^2+k^2 ψ= 0
withk^2 =^2 mE 2
ψ 1 =Aeikx+Be−ikx
Incident reflected atx= 0
Region 2, 0<x<L
d^2 ψ
dx^2−α^2 ψ= 0withα^2 =^2 m(W 2 −E)
ψ 2 =Ce−αx+Deαx
Region 3,x>L
d^2 ψ
dx^2+k^2 ψ= 0
withk^2 = 2 mE/^2
ψ 3 =Feikx
The second term is absent as there is no reflected wave coming from
right to left
The transmission coefficientT=|F|2
|A|^2
(c) Boundary conditions
ψ 1 (0)=ψ 2 (0)
dψ 1
dx∣
∣
∣
∣
x= 0=
dψ 2
dx∣
∣
∣
∣
x= 0
ψ 2 (L)=ψ 3 (L)
dψ 2
dx∣
∣
∣
∣
x=L=
dψ 3
dx∣
∣
∣
∣
x=L
(d) T= 16(E
W)(
1 −WE
)
e−^2 αLα^2 = 2 m(
W−E
^2
)
→α=√
2 mc^2 (W−E)
c=√
(2× 0. 511 ×(5−2)× 10 −^6
197. 3 × 10 −^15
= 8. 8748 × 109 m−^1