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PHYS 507
Homework III
Assigned:
October 30, 2003, Thursday.
Due:
November 10, 2003, Monday, at 5:00 pm.
1.
We have said in class that the “equation of motion” for the average position,
h
x
i
t
, of a
particle with the Hamiltonian
H
=
p
2
/
2
m
+
V
(
x
) is
m
d
2
dt
2
h
x
i
t
=
 h
V
0
(
x
)
i
t
,
and this is the classical equation of motion if
h
V
0
(
x
)
i
t
≈
V
0
(
h
x
i
t
).
Whenever this
approximation is good we are in the “classical limit” where use of quantum mechan
ics may be unnecessary.
If the approximation is bad then we have to use quantum
mechanics.
(a)
Consider a wave packet centered at
h
x
i
t
=
a
with a width of Δ
x
. By carrying out
a Taylor series expansion of
V
0
(
x
) around
a
, calculate the lowestorder term in the
diFerence between
h
V
0
(
x
)
i
t
and
V
0
(
h
x
i
t
). Calculate the percentage error made by
replacing quantum mechanical expression with a classical one. How large should
Δ
x
be in order to be in the classical limit?
(b)
Consider speci±cally the potential for a onedimensional Coulomb problem,
V
(
x
) =

A/

x

. When can we say that the classical solution for
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This note was uploaded on 02/11/2012 for the course MATH 435 taught by Professor Starg during the Spring '11 term at Al Ahliyya Amman University.
 Spring '11
 starg

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