Quiz 8
•
For the
n
th
harmonic oscillator energy eigenstate

n
!
, compute the position uncertainty
"
X.
–
Some useful equations:
(
)
†
2
A
A
X
+
=
!
2
2
Y
Y
Y
!
=
"
1
!
=
n
n
n
A
1
1
†
+
+
=
n
n
n
A
n
=
A
†
(
)
n
n
!
0
"
n
(
x
)
=
#
2
n
n
!
$
[
]
%
1/ 2
H
n
x
/
$
(
)
e
%
x
2
2
$
2
(
)
†
2
A
A
i
P
!
!
=
"
h
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Lecture 21: Heisenberg
Uncertainty Relation
PHY851 Quantum Mechanics I
Fall, 2008
M.G. Moore
Summary
•
The main results were:
1
!
=
n
n
n
A
1
1
†
+
+
=
n
n
n
A
0
0
=
A
n
=
A
†
(
)
n
n
!
0
"
0
(
x
)
=
#
$
[
]
%
1/ 2
e
%
x
2
2
$
2
1
0
†
=
A
"
1
(
x
)
=
2
#
$
[
]
%
1/ 2
2
x
$
e
%
x
2
2
$
2
"
n
(
x
)
=
#
2
n
n
!
$
[
]
%
1/ 2
H
n
x
/
$
(
)
e
%
x
2
2
$
2
)
(
x
x
n
X
x
n
!
=
)
(
1
)
(
2
)
(
2
1
x
n
n
x
x
n
x
n
n
n
!
!
!
!
=
"
"
#
"
Most efficient way
to compute high
orbitals numerically:
n
n
n
H
)
2
/
1
(
+
=
!
h
Heisenberg Uncertainty Relation
•
Most of us are familiar with the Heisenberg
Uncertainty relation between position and
momentum:
•
Are the similar relations between other
operators?
•
What are the consequences of Uncertainty
Relations?
2
h
!
"
"
P
X
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Variance
•
The uncertainties are also called ‘variances’
defined as
•
Why is this important?
•
Consider a distribution
P
(a)
,
–
The average of the distribution is:
•
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 Fall '08
 MichaelMoore
 mechanics, Energy, Uncertainty Principle, Heisenberg uncertainty relation

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