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Assignment #9
(Do not handin)
PHYS446
1. In this problem you will solve the isotropic simple harmonic oscillator problem using the power
series method. The potential is
1
2
m
2
r
2
for a particle of mass
m
.
(a) Write the Schrödinger equation for the function
u
r
=
r R
r
with angular momentum
l
.
(b) What are the asymptotic behaviors of
u
r
for large and small
r
? Call your result
f
r
in
the case of large
r
and write that
u
r
=
f
r
S
r
, where
S
r
is a power series in
r
,
S
r
=
∑
j
=
0
∞
b
j
r
j
. Find the differential equation satisfied by
S
r
by substituting your
expression for
u
r
into the Schrödinger equation.
(c) Substitute the power series into the equation you just found and derive the recursion relation
satisfied by the coefficients. Remember that this is done by demanding the equation be satisfied
order by order in
r
, and shifting the summation index to find the coefficient of each power of
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 Fall '08
 Vachon
 mechanics, Mass, Power

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