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Physics 137B, Fall 2007, Moore
Problem Set 9 Solutions
1.
Let
a
denote the hydrogen 1s state and let
b
denote the the 2p state with magnetic
quantum number
m
. In the notation

n`m
i
, we have

a
i
=

100
i
, and

b
i
=

21
m
i
.
The transition rate for spontaneous emission, calculated via the statistical argument
due to Einstein, is given by Eq. (11.76):
W
s
ab
=
ω
3
ba
3
πc
3
~
±
0

D
ba

2
.
(1)
Here
ω
ba
=
E
2

E
1
~
=

3
4
E
1
~
=
3
μα
2
c
2
8
~
(2)
and
D
ba
=
h
b
 
e
r

a
i
=

e
h
21
m

r

100
i
.
For a ﬁxed value of the magnetic quantum number
m
, the dipole moment
D
ba
can
be calculated most easily by expressing the vector
r
in terms of socalled
spherical
tensor operators
. To introduce this notion, ﬁrst deﬁne three new basis vectors
e
+1
=

1
√
2
(ˆ
x

i
ˆ
y
)
,
e

1
=
1
√
2
(ˆ
x
+
i
ˆ
y
)
,
e
0
= ˆ
z .
(The fact that we are using complex vectors to span real 3dimensional space looks
ﬁshy, but it turns out to be harmless. Note that these vectors are orthogonal in the
complex sense,
e
*
m
·
e
m
0
=
δ
m,m
0
.) Next we deﬁne new coordinates
r
m
≡
e
*
m
·
r
for
m
= 0
,
±
1. Explicitly
r
+1
=

1
√
2
(
x
+
iy
)
,
r

1
=
1
√
2
(
x

iy
)
,
r
0
=
z .
1
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View Full DocumentUsing the orthogonality property mentioned above, we can write

D
ba

2
=
e
2
(
h
b

r
+1

ψ
a
i
2
+
h
ψ
b

r

1

a
i
2
+
h
ψ
b

r
0

ψ
a
i
2
)
=
e
2
X
m
0
h
21
m

r
m
0

100
i
2
.
The motivation for these deﬁnitions is that they allow us to express
r
m
simply
in terms of the spherical harmonic functions that we all know and love. Looking at
Table 6.1, for instance, we see that with these deﬁnitions
r
m
=
r
4
π
3
rY
1
m
(
θ, φ
)
,
m
= 0
,
±
1
.
Now the hydrogen atom wavefunctions are
ψ
n`m
(
r, θ, φ
) =
R
n`
Y
`m
(
θ, φ
), so
h
21
m

r
m
0

100
i
=
Z
d
3
r
ψ
*
21
m
(
r
)
r
4
π
3
rY
1
m
0
(
θ, φ
)
ψ
100
(
r
)
=
r
4
π
3
Z
∞
0
r
3
R
21
(
r
)
R
10
(
r
)
dr
Z
Y
1
m
(
θ, φ
)
*
Y
1
m
0
(
θ, φ
)
Y
00
(
θ, φ
) sin
θ dθ dφ
=
1
√
3
δ
m,m
0
Z
∞
0
r
3
R
21
(
r
)
R
10
(
r
)
dr ,
since
Y
00
=
1
√
4
π
and the spherical harmonics have the orthogonality property
Z
Y
`m
(
θ, φ
)
*
Y
`
0
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 Fall '07
 MOORE
 Physics, mechanics

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