29.6 Explicit 2p to 1s Decay Rate
Starting from the summary equation for electric dipole transitions,above,
Γtot=
αω^3 in
2 πc^2∑
λ∫
dΩγ∣ ∣ ∣ ∣ ∣ ∣
√
4 π
3∫∞
0r^3 drR∗nnℓnRniℓi∫
dΩYℓ∗nmn(
ǫzY 10 +−ǫx+iǫy
√
2Y 11 +
ǫx+iǫy
√
2Y 1 − 1
)
Yℓimi∣ ∣ ∣ ∣ ∣ ∣
2we specialize to the 2p to 1s decay,
Γtot=
αω^3 in
2 πc^2∑
λ∫
dΩγ∣
∣
∣
∣∣
∣
√
4 π
3∫∞
0r^3 drR∗ 10 R 21∫
dΩY 00 ∗(
ǫzY 10 +−ǫx+iǫy
√
2Y 11 +
ǫx+iǫy
√
2Y 1 − 1
)
Y 1 mi∣
∣
∣
∣∣
∣
2perform the radial integration,
∫∞0r^3 drR∗ 10 R 21 =∫∞
0r^3 dr[
2
(
1
a 0) (^32)
e−r/a^0
][
1
√
24
(
1
a 0) (^52)
re−r/^2 a^0
]
=
1
√
6
(
1
a 0) 4 ∫∞
0r^4 dre−^3 r/^2 a^0=
1
√
6
(
1
a 0) 4 (
2 a 0
3) 5 ∫∞
0x^4 dxe−x=
1
√
6
(
2
3
) 5
a 0 (4!)= 4
√
6
(
2
3
) 5
a 0and perform the angular integration.
∫
dΩ Y 00 ∗
(
ǫzY 10 +−ǫx+iǫy
√
2Y 11 +
ǫx+iǫy
√
2Y 1 − 1
)
Y 1 mi=
1
√
4 π∫
dΩ(
ǫzY 10 +−ǫx+iǫy
√
2Y 11 +
ǫx+iǫy
√
2Y 1 − 1
)
Y 1 mi=
1
√
4 π(
ǫzδmi 0 +−ǫx+iǫy
√
2δmi(−1)+ǫx+iǫy
√
2δmi 1)
∣
∣
∣
∣
∫
dΩ Y 00 ∗(
ǫzY 10 +−ǫx+iǫy
√
2Y 11 +
ǫx+iǫy
√
2Y 1 − 1
)
Y 1 mi∣
∣
∣
∣
2=
1
4 π(
ǫ^2 zδmi 0 +1
2
(ǫ^2 x+ǫ^2 y)(δmi(−1)+δmi 1 ))
Lets assume the initial state is unpolarized, so we will sum overmiand divide by 3, the number of
differentmiallowed.
1
3∑
mi∣
∣
∣
∣
∫
dΩ Yℓ∗nmn(
ǫzY 10 +
−ǫx+iǫy
√
2Y 11 +
ǫx+iǫy
√
2Y 1 − 1
)
Yℓimi∣
∣
∣
∣
2