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Quantum Mechanics
Problem Sheet 1
(These problems will be discussed in the practice session on Friday, 20 Jan 2006)
1. Which of the following functions make good wave functions for describing a particle moving in one dimen
sion? Sketch the probability density for each of them and check whether it is normalizable. Determine the
normalization constant
C
n
for those that are normalizable. (The parameters
λ
n
are all real and positive.)
(a)
ψ
1
(
x
) =
0
for
x <
0
,
C
1
sin(
λ
1
x
)
for 0
≤
x
≤
2
π/λ
1
,
0
for
x >
2
π/λ
1
.
(b)
ψ
2
(
x
) =
C
2
1
x
+
λ
2
for all
x .
(c)
ψ
3
(
x
) =
C
3
exp(
λ
3
x
2
)
for all
x .
(d)
ψ
4
(
x
) =
C
4
exp(

λ
4

x

)
for all
x .
2. The wave function of a particle moving in one dimension is given by:
ψ
(
x
) =
±
B
exp(
βx
)
for
x <
0
B
exp(

2
βx
)
for
x
≥
0
,
where
β
is a real and positive constant.
(a) What is the probability density for Fnding the particle along the
x
axis? Sketch this function.
(b) Calculate the normalization constant
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This note was uploaded on 04/07/2008 for the course PHYS F3026 taught by Professor Eberlein during the Spring '06 term at Uni. Sussex.
 Spring '06
 EBERLEIN
 mechanics

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