Problem Set 1. (Fall 2006)
The charge density for a hydrogenic p state can be written as
r
,
2
3
e
a
0
3
64
r
a
0
2
exp
r
a
0
P
0
cos
P
2
cos
#
e
4.8
10
10
statCoulomb
,
a
0
0.529
10
8
cm
.
(a) Determine the multipole moments of this charge distribution and give the potential for
large
r
in terms of these moments. (b) Dtermine the potential, near the origin, correct to order
r
2
. (c) Determine the interaction energy with a nuclear quadrupole moment,
Q
10
24
cm
2
,
located at at the origin.
Solution:
Part a
:
The multipole expansion for the potential for
r
r
is in Gaussian units (see Jackson, Eq.
4.2):
r
,
,
l
,
m
4
2
1
r
r
,
,
Y
m
,
r
2
dr
d
Y
m
,
r
1
where the multipole moment of the charge distribution,
q
lm
, is given by:
q
lm
r
r
,
,
Y
m
,
r
2
dr
d
.
We can write this in a general way by first noting that the charge distribution is
r
,
,
2
3
e
a
0
3
64
u
2
exp
u
4
Y
00
,
4
5
Y
20
,
where
u
r
a
0
Thus the multipole moments are:
q
lm
2
3
a
0
e
64
u
4
exp
u
du
Y
m
,
4
Y
00
,
4
5
Y
20
,
d
.
2
3
a
0
e
64
4
!
4
0
m
0
4
5
2
m
0
Thus only
m
0 terms will be present, in particular only
Y
00
,
and
Y
20
,
.The
non-zero moments are
q
00
r
,
Y
00
,
r
2
drd
e
4
and
q
20
r
,
Y
20
,
r
2
drd
65
a
0
2
e
4
The potential for large
r
,
r
a
0
, is approximated by