130_notes.dvi

ADDITION OF ANGULAR MOMENTUM

J~=~L+S~ |ℓ−s|≤j≤ℓ+s L~·S=^1

2 (J

(^2) −L (^2) −S (^2) )
ψjmjℓs=

∑

mℓms

C(jmj;ℓmℓsms)Yℓmℓχsms=

∑

mℓms

〈jmjℓs|ℓmℓsms〉Yℓmℓχsms

ψj,mj=ψℓ+^12 ,m+^12 =

√

ℓ+m+1

2 ℓ+1 Yℓmχ++

√

ℓ−m

2 ℓ+1Yℓ,m+1χ− fors=

1

2 and anyℓ

ψj,mj=ψℓ− (^12) ,m+ 12 =

√

ℓ−m

2 ℓ+1Yℓmχ+−

√

ℓ+m+1

2 ℓ+1 Yℓ,m+1χ− fors=

1

2 and anyℓ

PERTURBATION THEORY AND RADIATIVE DECAYS

En(1)=〈φn|H 1 |φn〉 En(2)=

∑

k 6 =n

|〈φk|H 1 |φn〉|^2

E(0)n−Ek(0) c

(1)

nk=

〈φk|H 1 |φn〉

E(0)n −E(0)k

cn(t) =i^1 h ̄

∫t

0

dt′ei(En−Ei)t

′/ ̄h

〈φn|V(t′)|φi〉

Fermi’s Golden Rule: Γi→f=^2 h ̄π|〈ψf|V|ψi〉|^2 ρf(E)

Γi→f=^2 ̄hπ

∫ ∏

k

(V d

(^3) pk
(2π ̄h)^3 )|Mfi|
(^2) δ (^3) (momentum conservation)δ(Energy conservation)
Γradm→k= 2 πmα (^2) c 2

∫

dΩpωkm|〈φm|e−i

~k·~r

ˆǫ·~p|φk〉|^2

ΓEm^1 →k= 2 πcα 2

∫

dΩpω^3 km|〈φm|ǫˆ·~r|φk〉|^2 ∆l=±1, ∆s= 0

ˆǫ·rˆ=

√

4 π

3 (ǫzY^10 +

−ǫx√+iǫy

2 Y^11 +

ǫx√+iǫy

2 Y^1 −^1 ) ˆǫ·

~k= 0

I(ω)∝(ω−ω 0 Γ) 2 /+(Γ^2 /2) 2 Γcollision=Pσ

√

3

mkT (

∆ω

ω)Dopler=

√

kT

mc^2

(ddσΩ)BORN= 4 π^12 h ̄ 4 ppifmfmi|V ̃(∆ )~ |^2 V ̃(∆ ) =~

∫

d^3 ~r e−i

∆~·~r

V(~r) ∆ =~ ~pf− ̄h~pi

ATOMS AND MOLECULES

Hund: 1) maxs 2)maxℓ(allowed) 3) minj(≤^12 shell) else maxj

Erot=ℓ(ℓ+1) ̄h

2

2 I ≈

1

2000 eV Evib= (n+

1

2 ) ̄hω≈

1

50 eV