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PHY 4604 Spring 2012 – Homework 3
Due at the start of class on Wednesday, February 15.
No credit will be available for
homework submitted after the start of class on Monday, February 20.
Answer all three questions. Please write neatly and include your name on the front page of
your answers. You must also clearly identify all your collaborators on this assignment. To
gain maximum credit you should explain your reasoning and show all working.
Your may ﬁnd useful the following integrals:
Z
∞
0
x
2
n
e

x
2
dx
=
√
π
(2
n
)!
2
2
n
+1
n
!
Z
∞
0
x
2
n
+1
e

x
2
dx
=
n
!
2
1. For a particle of mass
m
moving in the potential
V
(
x
)=
1
2
mω
2
x
2
, it is often convenient
to express the position and momentum operators in terms of the ladder operators
a
±
:
x
=
r
~
2
mω
(
a
+
+
a

)
,p
=
i
r
~
mω
2
(
a
+

a

)
.
In answering this question, (i) it should not be necessary to perform any integrals over
x
, and (ii) you must not make use of the relation
h
p
i
=
md
h
x
i
/dt
.
(a) Evaluate
h
x
i
,
h
x
2
i
,
σ
x
,
h
p
i
,
h
p
2
i
,and
σ
p
for the stationary state
ψ
n
(where
n
=0
is the ground state). Check whether the uncertainty principle is obeyed.
(b) Evaluate
h
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This note was uploaded on 02/15/2012 for the course PHY 4604 taught by Professor Field during the Spring '07 term at University of Florida.
 Spring '07
 Field
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