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73. (a) The wave function is now given by
Ψ(
x, t
)=
ψ
0
h
e
i
(
kx
−
ωt
)
+
e
−
i
(
kx
+
)
i
=
ψ
0
e
−
iωt
(
e
ikx
+
e
−
ikx
)
.
Thus

Ψ(
x, t
)

2
=
¯
¯
ψ
0
e
−
iωt
(
e
ikx
+
e
−
ikx
)¯
¯
2
=
¯
¯
ψ
0
e
−
iωt
¯
¯
2
¯
¯
e
ikx
+
e
−
ikx
¯
¯
2
=
ψ
2
0
¯
¯
e
ikx
+
e
−
ikx
¯
¯
2
=
ψ
2
0

(cos
kx
+
i
sin
)+(cos
−
i
sin
)

2
=4
ψ
2
0
(cos
)
2
=2
ψ
2
0
(1 + cos 2
)
.
(b) Consider two plane matter waves, each with the same amplitude
ψ
0
/
√
2 and traveling in opposite
directions along the
x
axis. The combined wave Ψ is a standing wave:
Ψ(
x, t
ψ
0
e
i
(
kx
−
)
+
ψ
0
e
−
i
(
kx
+
)
=
ψ
0
(
e
ikx
+
e
−
ikx
)
e
−
iωt
=(
2
ψ
0
cos
)
e
−
iωt
.
Thus, the squared amplitude of the matter wave is

Ψ(
x, t
)

2
=(2
ψ
0
cos
)
2
¯
¯
e
−
iωt
¯
¯
2
ψ
2
0
(1 + cos 2
)
,
which is shown below.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2468
kx
(c) We set
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This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.
 Fall '08
 SPRUNGER
 Physics

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