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Lecture_10_-_Finite_Square_Well.pdf-Today (1) Review of time dependent
Page 29
29
wire
0
Energy
x
0
L
V(x)
Region I
Region II
Region III
)
(
)
(
)
(
)
(
2
2
2
2
x
E
x
x
V
dx
x
d
m
Need to solve Schrodinger Eqn:
4.7eV
E
electron
In Region II
total energy E > potential energy V
)
(
)
(
2
)
(
2
2
2
x
E
V
m
dx
x
d
Negative number
)
(
2
x
k
When E>V: Solutions = sin(kx), cos(kx), e
ikx
.
Always expect sinusoidal functions
k is real


Page 30
wire
0
Energy
x
0
L
V(x)
Region I
Region II
Region III
)
(
)
(
)
(
)
(
2
2
2
2
x
E
x
x
V
dx
x
d
m
Need to solve Schrodinger Eqn:
4.7eV
E
electron
In Region III
total energy E < potential energy V
)
(
)
(
2
)
(
2
2
2
x
E
V
m
dx
x
d
Positive
)
(
2
x
is real
What functional forms of
(x)
work?
a. e
i
x
b. sin(
x)
c. e
x
d. more than one of these
In class activity
30


Page 31
Answer is C: e
x
… could also be e
-
x
.
Exponential decay or growth
Why not e
i
x
?
)
(
)
(
2
2
x
x
LHS
RHS
wire
0
Energy
x
0
L
V(x)
Region I
Region II
Region III
)
(
)
(
)
(
)
(
2
2
2
2
x
E
x
x
V
dx
x
d
m
Need to solve Schrodinger Eqn:
4.7eV
E
electron
In Region III
total energy E < potential energy V
)
(
)
(
2
)
(
2
2
2
x
E
V
m
dx
x
d
Positive
)
(
2
x
is real
What functional forms of
(x)
work?
In class activity
31


Page 32
32
wire
0
Energy
x
0
L
V(x)
Region I
Region II
Region III
4.7eV
E
electron
x
x
III
Be
Ae
x
)
(
)
cos(
)
sin(
)
(
kx
D
kx
C
x
II
x
x
I
Fe
Ee
x
)
(
What will wave function in Region III look like?
What makes sense for constants A and B?
a. A must be 0
b. B must be 0
c. A and B must be equal
d. A=0 and B=0
e. A and B can be anything, need more info.
In class activity


Page 33
33
wire
0
Energy
x
0
L
V(x)
Region I
Region II
Region III
4.7eV
E
electron
x
x
III
Be
Ae
x
)
(
x
x
I
Fe
Ee
x
)
(
What will wave function in Region III look like?
What makes sense for constants A and B?
Answer is a.
A must be 0 .. otherwise
blows up as x gets bigger.
This doesn’t make sense!
and probability should
0 at large x!
Need to be able to normalize
)
cos(
)
sin(
)
(
kx
D
kx
C
x
II


Page 34
34
V=0 eV
0
L
4.7 eV
Energy
x
E
electron
x
III
Be
x
)
(
Inside well (E>V):
(Region II)
)
(
)
(
2
2
2
x
k
dx
x
d
II
II
)
(
)
(
2
2
2
x
dx
x
d
III
III
Outside well (E<V):
(Region III)
Boundary
Conditions:
continuous
)
(
L
continuous
)
(
dx
L
d


x
as
0
)
cos(
)
sin(
)
(
kx
D
kx
C
x
II
Outside well
(E<V):
(Region I)
On HW
next week
)
(
)
(
L
L
III
II
dx
L
d
dx
L
d
III
II
)
(
)
(


Page 35
35
V=0 eV
0
L
4.7 eV
Energy
x
E
electron
)
(
)
(
2
)
(
2
2
2
x
E
V
m
dx
x
d
Inside well (E>V):
Outside well (E<V):
Electron is delocalized … spread out.
Some small part of wave is where
Total energy is less than potential
energy!
“Classically forbidden”
region.


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Santa Clara University
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