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PHY4604
R. D. Field
Department of Physics
Chapter2_4.doc
University of Florida
The Infinite Square Well (2)
Normalized Wavefunctions:
Now we require that
1
)
(
)
(
=
∫
∞
∞
−
∗
dx
x
x
n
n
ψ
,
and hence
1
2
sin
)
/
(
sin
2
0
2
2
0
2
2
=
=
=
∫
∫
LA
d
n
LA
dx
L
x
n
A
n
L
π
θ
.
Thus ,
L
A
2
/
1
=
and
h
/
)
(
)
,
(
t
iE
n
n
n
e
x
t
x
−
=
Ψ
with
)
/
sin(
2
)
(
L
x
n
L
x
n
=
.
Probability Density:
The probability
density for the n
th
state is
)
/
(
sin
2

)
,
(

)
(
2
2
L
x
n
L
t
x
x
n
n
ρ
=
Ψ
=
.
Note that
ρ
n
and not a function of time!
These states are
“stationary states”
(
i.e.
independent of time
).
Average Value of x:
The average
position for the n
th
state is
2
4
2
)
(
sin
2
)
/
(
sin
2
)
(
)
(
2
2
2
2
0
2
2
2
0
2
*
L
n
n
L
d
n
L
dx
L
x
n
x
L
dx
x
x
x
x
n
L
n
=
=
=
=
Ψ
Ψ
=
>
<
∫
∫
∫
∞
∞
−
Average Value of x
2
:
The average value of x
2
in the n
th
state is
−
=
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.
 Spring '07
 FIELDS
 mechanics

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