Today’s class:
Quantum tunneling
4.7 eV
y
x
III
Be
x
α
ψ
−
=
)
(
Inside well (E>V):
(Region II)
)
(
)
(
2
2
2
x
k
dx
x
d
II
II
ψ
ψ
−
=
)
(
)
(
2
2
2
x
dx
x
d
III
III
ψ
α
ψ
=
Outside well (E<V):
(Region III)
)
cos(
)
sin(
)
(
kx
D
kx
C
x
II
+
=
ψ
Outside well
(E<V):
(Region I)
Review:
The finite square well
V=0 eV
0
L
Energy
x
E
electron
1)
ψ
(x)
has to be continuous:
2)
ψ
(x)
has to be smooth:
3)
ψ
(x
∞
)
0 (required for normalization)
)
(
)
(
L
L
III
II
ψ
ψ
=
)
(
'
)
(
'
L
L
III
II
ψ
ψ
=
0
L
E
electron
Review:
‘Penetration depth’
2
m
−
=
α
E<V: Classically forbidden region
)
(
L
x
−
−
=
α
ψ
)
(
2
E
V
h
)
(
Be
x
, with
)
(
L
ψ
e
L
/
)
(
ψ
1/
α
Penetration depth:
Distance 1/
α
over which the wave function
decays to 1/e of its initial value at
the potential boundary (here
ψ
(L)
):
For VE = 4 eV, 1/
α
~ 0.1 nm (very small ~ an atom!!!)
x
ψ
0
L
x
Be
x
α
ψ
−
=
)
(
)
(
2
2
E
V
m
−
=
h
α
What changes would increase how far the wave
penetrates into the classically forbidden region?
with:
Q1
A. Decrease potential depth (= workfunction of metal)
B. Increase potential depth
C. Decrease wire length L
D. Increase wire length L
E. More than one of the above
Wide vs. narrow finite potential well
2
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 Spring '08
 staff
 Physics, 2m, 1 L, Ψ, 0 l, 0.1 nm

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