University of Colorado, Department of Physics
PHYS3220, Fall 09, HW#11
due Wed, Nov 4, 2PM at start of class
1. Analytic solution of the harmonic oscillator (Total: 20 pts)
In this problem we go through the analytic solution of the timeindependent Schr¨
odinger
equation for the harmonic oscillator

~
2
2
m
d
2
χ
(
x
)
dx
2
+
1
2
mω
2
x
2
χ
(
x
) =
Eχ
(
x
)
(1)
a) It is convenient to simplify the problem by introducing the two dimensionless variables
ξ
=
p
mω
~
x
and
K
=
2
E
~
ω
Show that the timeindependent Schr¨
odinger equation can
be written as
d
2
χ
(
ξ
)
dξ
2
=
(
ξ
2

K
)
χ
(
ξ
)
b) Show that in the limit

x
 → ∞
the above equation can be approximated as
d
2
χ
(
ξ
)
dξ
2
≈
ξ
2
χ
(
ξ
)
and an approximate solution is given by
χ
(
ξ
)
≈
A
exp(

ξ
2
/
2) +
B
exp(+
ξ
2
/
2)
What is the constraint needed to be sure that
χ
(
ξ
) can be normalized? (Don’t actually
try to normalize the function.)
c) The original equation turns out to simplify if we extract the asymptotic (large

x

)
behavior of
χ
and solve for what’s left. Accordingly, we define
H
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 Fall '08
 STEVEPOLLOCK
 Physics, timeindependent Schr¨dinger equation

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