PHYS 402 Homework---Due February 14
Griffiths 4.18
Griffiths 4.20 (parts a,c,d)
1. The spherical harmonics are a complete orthonormal set describing smooth angular functions.
Completeness means than any smooth angular function may be written in the form
),(),(
,,
φϑφϑ
∑
=
ml
m
lml Ycf . Orthonormailty implies 2
,
2
,),(
∫
∑Ω=
φϑ
fdc
ml ml . Find the coefficients
ml
c, for the function
()
)cos()2sin()sin(),(
φϑϑφϑ
+=f.
2. Explicitly show that ),(
2
2
φϑ
Y,),(
1
2
φϑ
Y and ),(
0
2
φϑ
Yfrom table 4.2 of Griffiths satisfy the angular
equation (4.18), are normalized and are orthogonal to each other.
3. Find the following matrix elements for states labeled ml,:
a. 2,22,1 z
L
b. 2,22,2 z
L
c. 2,22,2 x
L
d. 2,21,2 x
L
e. 2,21,2 −
x
L
f. 2,21,2 −− x
L
g. 2,21,2 y
L
