PHY4221 Quantum Mechanics I (Fall 2004)
Quiz 2 (Answer)
Step 1:
[
]
[
]
(
29
[
]
1
1
1
1
1
2
1
2
2
1
2
2
2
1
2
2
1
1
1
,
,
,
,
,
,
,
,
,
,
2
,
,
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
x p
x pp
p
x p
x p p
p
x p
i p
p
x pp
i p
p
p x p
x p p
i p
p
x p
pi p
i p
p
x p
i p
p
x p
ni p
p
x I
ni p
ni p


















=
=
+
=
+
=
+
=
+
+
=
+
+
=
+
= ×××
=
+
=
+
=
h
h
h
h
h
h
h
h
h
1
n

.
Step 2:
In order to be able to prove the second equation using the result of the step 1, it is
clearly that you should expand
(
29
F
p
using the series of
n
p
. It is again a math work.
You may expect to naturally using the Taylor expansion around
p
=0 as following:
(
29
(
29
(
29
(
29
(
29
(
29
2
0
0
0
0
2!
0
!
n
n
n
F
F
p
F
F
p
p
F
p
n
∞
=
′′
′
=
+
+
+×××
=
∑
.
In the above the superscript
(
n
)
denotes the
n
order derivative of
F
(
p
) with respect to
p
.
Note that although the Taylor expansion above itself may not converge, you, with
your knowledge of mathematical physics, would consider it valid in the form, which
really matters. So there is no problem with this expansion.
Now substitute the expansion into the left hand of the tobeproved equation:
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