Chem 3890
Physical Chemistry I
Fall 2009
Chemistry 3890
Problem Set 4
Fall 2009
Due: in class, Friday, October 9
Last revised:
October 4, 2009
Additions/corrections:
✑
Wavefunctions live in Hilbert space.
D.J. Griffiths,
Quantum mechanics
, p94
Quantum phenomena do not occur in a Hilbert space; they occur in a laboratory.
Asher Peres
Reminder
: For the Mathematica portions of the HW, you must submit a printout of an
Evaluated
notebook. It
☞
is a good idea to first
Save
your evaluated notebook in pdf format, then print the pdf file. Printouts of unevaluated
notebooks will not be graded.
Problem 1
In this problem we solve the timeindependent Schr¨odinger equation for a particle in a box that
has a step at the midpoint of the box. The Schr¨odinger equation is

planckover2pi1
2
2
m
d
2
ψ
(
x
)
d
x
2
+
V
(
x
)
ψ
(
x
) =
Eψ
(
x
)
,
(1)
where
m
is the particle mass,
E
is a real parameter (the energy), and
V
(
x
) is the potential
V
(
x
) =
∞
x<
0
0
0
≤
x
≤
a
2
¯
V
a
2
≤
x
≤
a
∞
x>a
(2)
1. Make a sketch of this potential for
¯
V >
0.
2. Show
that the general solution for
ψ
(
x
) on the interval 0
≤
x
≤
a/
2 can be written as
ψ
(
x
) =
A
1
sin(
k
1
x
) +
B
1
cos(
k
1
x
)
,
(3)
with
k
1
≡
√
2
mE/
planckover2pi1
. What is the boundary condition that
ψ
(
x
) satisfies at
x
= 0? Impose this BC on
ψ
(
x
).
3. There is a corresponding expression for
ψ
(
x
) on the interval
a/
2
≤
x
≤
a
,
ψ
(
x
) =
A
2
sin(
k
2
x
) +
B
2
cos(
k
2
x
)
,
(4)
with
k
2
≡
radicalbig
2
m
(
E

¯
V
)
/
planckover2pi1
. (We take 0
≤
¯
V <E
.) What is the BC that
ψ
(
x
) must satisfy at
x
=
a
?
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 '08
 HINES, M
 Physical chemistry, pH, Energy, appropriate matching conditions

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