EP 471 – Homework Set #3
1)
Legendre’s differential equation,
0
,
()
22
12
xy x
y
y
λ
′′
′
−
−+=
arises in numerous physical problems, particularly in BVPs involving spherical
symmetry. In the quantum theory of the atom this is related to the polar part of the
electron wave function. In these problems, the amplitude of the wave function is
related to the probability of finding the electron. If the potential is symmetric
(Coulomb attraction), the wave amplitude
( )
2
x
ψ
has to be symmetric, leading to one
of two kinds of solutions:
Even:
() ( )
x
x
=−
,
Odd:
( ) ( )
x
x
=
−−
In terms of the current problem, we wish to solve the above equation over the interval
1
x
subject to one of two sets of boundary conditions:
Even:
11
;
1
−≤ ≤+
y
−=
( )
y
+
=
;
( )
y
′ +
=
Odd :
;
1
y
1
−
( )
y
+
=
;
( )
y
′ +
=
To solve this problem numerically, we should first write it as:
0
where
ε
is a small number (suggestion: use
6
10
y
y
ελ
′
−
+
=
+−
−
=
) so that we don’t get a divideby
zero error at the ends of the interval. As a check on the result, you may be aware that
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 Spring '08
 Witt
 Numerical Analysis, finite difference, Finite difference method, Boundary conditions, Iη, EC1 Iη

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