Quiz #4: 1D Schr¨
odinger Eqn
Friday May 7
11:00am to 11:50am
[Note:
This is a closedbook exam.
No calculator is allowed.
The total score
of this exam is 50 points.
Points assigned to each problem is a guide to how
many minutes you should spend working on it.]
Possibly useful formulae:
•
timedependent Schr¨
odinger’s equation:
i
¯
h
∂
∂t
Ψ =

¯
h
2
2
m
∂
2
∂x
2
Ψ +
U
(
x
)Ψ
•
timeindependent Schr¨
odinger’s equation:

¯
h
2
2
m
∂
2
∂x
2
ψ
+
U
(
x
)
ψ
(
x
) =
Eψ
(
x
)
•
expectation values:
h
x
m
i
=
R
∞
∞
dxx
m
P
(
x
)
•
standard deviation: Δ
x
=
q
h
x
2
i  h
x
i
2
, Δ
p
=
q
h
p
2
i  h
p
i
2
•
energy levels of the harmonic oscillator:
E
n
= (
n
+
1
2
)¯
hω
c
for
n
= 0
,
1
,
2
,...
, with
ω
c
being the
frequency of the classical oscillator.
•
sin(
θ
) = 0 for
θ
= 0
,
±
π,
±
2
π,
3
π,.
..
•
cos(
θ
) = 0 for
θ
=
±
π/
2
,
±
3
π/
2
,
±
5
π/
2
,...
•
R
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 Spring '08
 Hirsch
 Physics, Standard Deviation, wave function, expectation values

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