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Chem 120A
Square Potentials
02/24/06
Spring 2006
Lecture 16
READING:
Engel: Chapters 4 and 5
See ’McQuarrie square potentials’ supplement in Lecture Notes folder on blackboard.berkeley.edu
Last time we began to look at the behavior of a quantum particle incident upon a square potential barrier
(see Figure 1). The wavefunctions in the three regions of the problem are given by:
Region I:
Ψ
I
(
x
) =
Ae
ikx
+
Be

ikx
Region III:
Ψ
III
(
x
) =
Qe
ikx
+
Fe

ikx
where
k
2
=
2
m
¯
h
2
(
E
)
Region II
E
<
V
:
Ψ
II
(
x
) =
Ce

κ
x
+
De
κ
x
where
κ
2
=
2
m
¯
h
2
(
V

E
)
E
>
V
:
Ψ
II
(
x
) =
C
0
e
ik
0
x
+
D
0
e

ik
0
x
where
k
2
0
=
2
m
¯
h
2
(
E

V
)
We can solve for the coefﬁcients (A, B, C, D, Q) by matching
Ψ
0
s
and their ﬁrst derivatives at the bound
aries. Note that terms with
e
+
i
α
x
(where
α
=
k
,
k
0
,
i
κ
) represent wavepackets moving to the right in each
region. Likewise terms with
e

i
α
x
represent wavepackets moving to the left. Classically if a particle hits an
energy barrier with
E
<
V
it will not go through the barrier, but will be reﬂected (T = 0 and R = 1 where
T is the transmission coefﬁcient and R is the reﬂection coefﬁcient). When
E
>
V
, the classical particle
will be transmitted through the barrier (T = 1 and R = 0). In the quantum mechanical case, very different
behavior is observed. The particle can still penetrate the barrier, even for
E
<
V
. In addition a quantum
particle with
E
>
V
is not perfectly transmitted, except at certain resonance conditions. To investigate this
behavior, we will solve for the transmision, T, and reﬂection, R, coefﬁcients. These were deﬁned last time as:
Chem 120A, Spring 2006, Lecture 16
1
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View Full Document T
=
±
±
±
±
Q
A
±
±
±
±
2
R
=
±
±
±
±
B
A
±
±
±
±
2
(1)
The particle is either transmitted or reﬂected, so their sum is unity
R
+
T
=
1
For
E
>
V
the transmission coefﬁcient is
1
T
=
1
+
1
4
V
2
E
(
E

V
)
sin
2
(
k
0
a
)
(2)
For
E
<
V
the transmission coefﬁcent is
T
=
1
1
+
1
4
V
2
E
(
V

E
)
sinh
2
(
κ
a
)
(3)
(you can follow the derivation of this equation in the ’McQuarrie square potentials’ supplement in Lecture
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This note was uploaded on 09/29/2009 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at University of California, Berkeley.
 Spring '07
 Whaley
 Physical chemistry, pH

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