Portal IFGWcabrera/teaching/aula 20... · 2010-06-13 · 32. Clebsch-Gordan coe cients 1...
Transcript of Portal IFGWcabrera/teaching/aula 20... · 2010-06-13 · 32. Clebsch-Gordan coe cients 1...
Time-Independent Perturbation Theory Perturbation WHH ˆˆˆ
0 += 0W H
while nnn EH ϕϕ 00
ˆ = nnn EH ψψ =ˆ
First Order nnnn WEE ϕϕ ˆ0 +=
−+= ∑
≠nkk
kn
nknn EE
WN ϕ
ϕϕϕψ 00
ˆ
Second Order 2
00 0
ˆˆ k n
n n n nk n n k
WE E W
E E
ϕ ϕϕ ϕ
≠
= + +−
∑
Degenerate states diagonalize the perturbation in each state’s degeneracy subspaces, one by one. If the Operator of the degeneracy commutes with the perturbation than the perturbation is diagonal & Perturbation theory gives exact results.
Scattering Theory
Radial Equation ( ) ( ) ( ) ( )2 2 2
2 2
1
2 2
d l lV r u r E u r
m dr mr
+− + + =
Boundary condition ( )0 0u =
Solution for free particle ( ) ( )22 ,kl lmj kr Yπ θ φΨ =
Particle current ( ) ( )Im2
jmi m
ψ ψ ψ ψ ψ ψ∗ ∗ ∗≡ ∇ − ∇ = ∇
Bessel function ( ) ( ) ( ) sin11 ll l d xl x dx xj x x= −
( ) ( )( ) ( )
12
12 1 !!0
lim sin
lim
l xxl
l lx
j x x l
j x x
π→∞
+→
= −
=
Free Wave expansion ( ) ( ) ( )0
2 1 cosikz ll l
l
e i l j kr P θ∞
=
= +∑Partial Wave appr. ( ) ( )2lim sinkl lr
u r kr l π δ→∞
= − +
Limiting condition ( ) 01l l kr+ >
where 0r is the potential effective distance
Scattering Amp. ( ) ( ) ( )0
12 1 sin cosli
k l ll
f l e Pk
δθ δ θ∞
=
= + ⋅∑
Differential Cross section ( ) 2d d fσ θΩ =
Total Cross section ( ) 22
0
4 2 1 sintot ll
lkπσ δ
∞
=
= +∑
Born Approximation ( ) ( ) 322
iq rmf V r e d rθπ
− ⋅= − ∫
where 2 2 sinf iq k k q k θ= − =
Central Pot. ( ) ( ) ( )20
2sin
mf r V r qr dr
qθ
∞
= − ⋅∫
condition ( )( )22
0
1 1ikrm V r e drk
∞
−∫
Time Dependent Perturbation Theory
Transition probability ( )2
20
1 fi
ti t
fi fiP V t e dtω ′′ ′= ∫
where ( ) ( )1 | , |fi f i fi f iE E V V r tω ϕ ϕ= − =
Conditions fi f iV E E− I-order: 1fiPAdiabatic Theorem short perturbations are felt like
delta functions, while slowly changing perturbation will not follow with transition.
Sinusoidal Perturbation ( )
( )22
2
22
2
sin
4
fi
fi
fifi
tVP
ω ω
ω ω
−
−≅
Conditions 1 fi
fit V tω
Fermi’s Golden Rule ( )22fi fi fR V Eπ ρ=
where ( )fEρ is energy density of final state Atomic Transitions Electric Dipole sine
DE zmV p tεω ω= −
Magnetic Dipole ( )2 2 coseDM x xmcV L S tε ω= − +
Electric Quadrupole ( )2 coseQE z ymcV yp zp tε ω= − +
Selection Rules The Integral
1 1 2 2 3 30l m l m l mY Y Y d∗ Ω ≠∫ only if
1) 1 2 3m m m= + 2) triangle can be created from 1 2 3, ,l l l 3) parity: 1 2 3l l l even+ − = Useful Relations for field polarization calculus
[ ] [ ]8 81 111 1 1 11 1 12 3 2 3 ix Y Y r y Y Y rπ π
− −= − − = − +
4103z r Yπ= ⋅ [ ],mip H r=
Useful Relations ( ) ( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
,
, 1 ,
, 1 ,
1
, 1 ,
llm lm
l mlm lm
ml m lm
mlm lm
Y Y
Y Y
Y Y
Y Y
π θ φ π θ φ
π θ φ θ φ
θ φ π θ φ
+
∗−
− + = −
− = −
= −
+ = −
( )2 2 2 3 10 0 2
1 1 1 1| | | | r a n r a n lklm klm klm klm
+= =
( )( )3 3 3 10 2
1 11
| |r a n l l l
klm klm+ +
=
2
00 20
11 21
2 2
10 22
1 5 3 1cos
4 2 24
3 15sin sin cos
8 8
3 1 15cos sin
4 4 2
i i
i
Y Y
Y e Y e
Y Y e
φ φ
φ
θππ
θ θ θπ π
θ θπ π
= = −
= − = −
= =
Angular Momentum Rotation Operator ( ) ( )ˆ
ˆ ˆexp inR L nα α= − ⋅
Orbital Angular Momentum L r p= × x z y y x z z y xL yp p z L zp p x L xp p y= − = − = − cos
sintanxL
i
ϕϕ
θ θ ϕ
∂ ∂= − −
∂ ∂
sincos
tanyLi
ϕϕ
θ θ ϕ
∂ ∂= −
∂ ∂
zLi ϕ
∂=
∂
2 22 2
2 2 2
1 1
tan sinL
θ θ θ θ ϕ
∂ ∂ ∂= − + +
∂ ∂ ∂
Spin operator 2S σ=
Pauli matrices 0 1 0 1 0
1 0 0 0 1x y z
i
iσ σ σ
−= = =
−
prop. , 2i j ijk kiσ σ ε σ = , 2i j ijσ σ δ=
kijkijji i σεδσσ +=
Rotation. prop. 2 ˆ2 2ˆcos sini ne i n
ασ α ασ− ⋅ = − ⋅
Ladder Operators x yJ J iJ± = ±
2 2z zJ J J J J− += + +
, ( 1) ( 1) , 1J j m j j m m j m± = + − ± ±
Commutation relations
,i j ijk kJ J i Jε = ⋅ 2, 0J J =
[ ],zJ J J± ±= ± 2 , 0J J± =
Spin addition 1 2J J J= +
( )1 2 1 2 1 2 J j j j j M m m= − + = +…
relations 2 2 21 2 1 22J J J J J= + +
2 2 21 2 1 2 1 2 1 22 z zJ J J J J J J J J+ − − += + + + +
( )11 2 1 2 1 2 1 22z zJ J J J J J J J+ − − +⋅ = + +
Spin states representation 1 1 1 1 1 1 1 12 2 2 2 2 2 22 1
1 1 1 1 1 1 1 12 2 2 2 2 2 22 1
, , ,
, , ,
l
l
l m l m m l m m
l m l m m l m m
+
+
+ = + + − + − + + −
− = + + + − − − + −
Spinors
( )
( )
12
12
12
12
12
12
12 ,
, ,12 ,
12 ,
, ,12 ,
12 1
12 1
l m
klk l m
l m
l m
klk l m
l m
l m YR r
l l m Y
l m YR r
l l m Y
−
+
+
−
+
+
+ +Ψ =
+ − +
− − +Ψ =
+ + +
Interaction Hamiltonians + Corrections
Spin-Orbit Coupling ( )2 2
12SO
V reH L Sm c r r
∂= ⋅
∂
Hydrogen 2
2 2 3
12SO
eH L Sc rµ
= ⋅
Correction ( )
( )( )
12
12
1 14 22
3 1 121
14
j j
nlj j
j lE mc
n j lα +
−+ +
= +∆ = = −
Weakly Relativistic correction
2 4
3 22 8Kp pEm m c
≅ − ( )202
12mvH H V
mc= − −
Correction 4
23 1
2
1 3 24 2nlE mc
n n lα
∆ = − +
Electromagnetic interaction ( )21
2q
EM m cH p A qϕ= − +
first order ( ) ( )2ˆ2 2qB
B LmcH L S L S Bω= − + = + ⋅
Larmor frequency 2qB
L mcω = − Correction ( )1 1
2 1 21 l J L lE M J lω +∆ = ± = ± for weak fields B SOH H
Identical Particles Permutation Operator 21 1 2 2 11 ;2 1 ;2P ϕ ϕ ϕ ϕ=
prop. † 221 21 21 1P P P= = eigenvalues: 1±
Tensor multiplication 1 ;2 |1 ;2 1 |1 2 | 2a b c d a c b d=
Symmetrizer 1ˆ!
S PN α
α
= ∑
two particles ( )1212
ˆ 1S P= + (normalized) Anti-Symmetrizer
1 even permutation1ˆ
1 odd permutation!A P
N α α αα
ε ε= =−
∑
two particles ( )1212
ˆ 1A P= − (normalized)
Proprieties † 2 † 2 0S S S A A A AS SA= = = = = =Symmetrization postulate a physical system of identical particles can be either completely symmetric or completely anti-symmetric.
Hydrogen Atom Fine structure constant 2 1
137e
cα = ≅
Bohr radius 2
20 mcea αµ
= =
Energy levels 22 21 1
2n nE mc α= −
Radial Functions 0 ,( ) ( )r
l nan lR r N r P re−= ⋅
e ar
aR 02
3
00,1 2 −−= ( ) ( )32 2 0
02,0 0 22 2 1rar
aR a e− −= −
( )3
2 2 0
0
12,1 03
2rar
aR a e− −= 01, 1
rnan
n nR Cr e−−− =
32. Clebsch-Gordan coefficients 1
32. CLEBSCH-GORDAN COEFFICIENTS, SPHERICAL HARMONICS,
AND d FUNCTIONS
Note: A square-root sign is to be understood over every coefficient, e.g., for −8/15 read −√
8/15.
Y 01 =
√3
4πcos θ
Y 11 = −
√3
8πsin θ eiφ
Y 02 =
√5
4π
(32
cos2 θ − 12
)Y 1
2 = −√
158π
sin θ cos θ eiφ
Y 22 =
14
√152π
sin2 θ e2iφ
Y −m` = (−1)mYm∗` 〈j1j2m1m2|j1j2JM〉= (−1)J−j1−j2〈j2j1m2m1|j2j1JM〉d `m,0 =
√4π
2`+ 1Ym` e−imφ
djm′,m = (−1)m−m
′djm,m′ = d
j−m,−m′ d 1
0,0 = cos θ d1/21/2,1/2
= cosθ
2
d1/21/2,−1/2
= − sinθ
2
d 11,1 =
1 + cos θ2
d 11,0 = − sin θ√
2
d 11,−1 =
1− cos θ2
d3/23/2,3/2
=1 + cos θ
2cos
θ
2
d3/23/2,1/2
= −√
31 + cos θ
2sin
θ
2
d3/23/2,−1/2
=√
31− cos θ
2cos
θ
2
d3/23/2,−3/2
= −1− cos θ2
sinθ
2
d3/21/2,1/2
=3 cos θ − 1
2cos
θ
2
d3/21/2,−1/2
= −3 cos θ + 12
sinθ
2
d 22,2 =
(1 + cos θ2
)2
d 22,1 = −1 + cos θ
2sin θ
d 22,0 =
√6
4sin2 θ
d 22,−1 = −1− cos θ
2sin θ
d 22,−2 =
(1− cos θ2
)2
d 21,1 =
1 + cos θ2
(2 cos θ − 1)
d 21,0 = −
√32
sin θ cos θ
d 21,−1 =
1− cos θ2
(2 cos θ + 1) d 20,0 =
(32
cos2 θ − 12
)
+1
5/25/23/2
3/2+3/2
1/54/5
4/5−1/5
5/2
5/2−1/23/52/5
−1−2
3/2−1/22/5 5/2 3/2
−3/2−3/24/51/5 −4/5
1/5
−1/2−2 1
−5/25/2
−3/5−1/2+1/2
+1 −1/2 2/5 3/5−2/5−1/2
2+2
+3/2+3/2
5/2+5/2 5/2
5/2 3/2 1/2
1/2−1/3
−1
+10
1/6
+1/2
+1/2−1/2−3/2
+1/22/5
1/15−8/15
+1/21/10
3/103/5 5/2 3/2 1/2
−1/21/6
−1/3 5/2
5/2−5/2
1
3/2−3/2
−3/52/5
−3/2
−3/2
3/52/5
1/2
−1
−1
0
−1/28/15
−1/15−2/5
−1/2−3/2
−1/23/103/5
1/10
+3/2
+3/2+1/2−1/2
+3/2+1/2
+2 +1+2+1
0+1
2/53/5
3/2
3/5−2/5
−1
+10
+3/21+1+3
+1
1
0
3
1/3
+2
2/3
2
3/23/2
1/32/3
+1/2
0−1
1/2+1/22/3
−1/3
−1/2+1/2
1
+1 1
0
1/21/2
−1/2
0
0
1/2
−1/2
1
1
−1−1/2
1
1
−1/2+1/2
+1/2 +1/2+1/2−1/2
−1/2+1/2 −1/2
−1
3/2
2/3 3/2−3/2
1
1/3
−1/2
−1/2
1/2
1/3−2/3
+1 +1/2+10
+3/2
2/3 3
3
3
3
3
1−1−2−3
2/31/3
−22
1/3−2/3
−2
0−1−2
−10
+1
−1
6/158/151/15
2−1
−1−2
−10
1/2−1/6−1/3
1−1
1/10−3/10
3/5
020
10
3/10−2/53/10
01/2
−1/2
1/5
1/53/5
+1
+1
−10 0
−1
+1
1/158/156/15
2
+2 2+1
1/21/2
1
1/2 20
1/6
1/62/3
1
1/2
−1/2
0
0 2
2−21−1−1
1−11/2
−1/2
−11/21/2
00
0−1
1/3
1/3−1/3
−1/2
+1
−1
−10
+100
+1−1
2
1
00 +1
+1+1
+11/31/6
−1/2
1+13/5
−3/101/10
−1/3−10+1
0
+2
+1
+2
3
+3/2
+1/2 +11/4 2
2
−11
2
−21
−11/4
−1/2
1/2
1/2
−1/2 −1/2+1/2−3/2
−3/2
1/2
1003/4
+1/2−1/2 −1/2
2+13/4
3/4
−3/41/4
−1/2+1/2
−1/4
1
+1/2−1/2+1/2
1
+1/2
3/5
0−1
+1/20
+1/23/2
+1/2
+5/2
+2 −1/21/2+2
+1 +1/2
1
2×1/2
3/2×1/2
3/2×12×1
1×1/2
1/2×1/2
1×1
Notation:J J
M M
...
. . .
.
.
.
.
.
.
m1 m2
m1 m2 Coefficients
−1/52
2/7
2/7−3/7
3
1/2
−1/2−1−2
−2−1
0 4
1/21/2
−33
1/2−1/2
−2 1
−44
−2
1/5
−27/70
+1/2
7/2+7/2 7/2
+5/23/74/7
+2+10
1
+2+1
41
4
4+23/14
3/144/7
+21/2
−1/20
+2
−1012
+2+10
−1
3 2
4
1/14
1/14
3/73/7
+13
1/5−1/5
3/10
−3/10
+12
+2+10
−1−2
−2−1012
3/7
3/7
−1/14−1/14
+11
4 3 2
2/7
2/7
−2/71/14
1/14 4
1/14
1/143/73/7
3
3/10
−3/10
1/5−1/5
−1−2
−2−10
0−1−2
−101
+10
−1−2
−12
4
3/14
3/144/7
−2 −2 −2
3/7
3/7
−1/14−1/14
−11
1/5−3/103/10
−1
1 00
1/70
1/70
8/3518/358/35
0
1/10
−1/10
2/5
−2/50
0 0
0
2/5
−2/5
−1/10
1/10
0
1/5
1/5−1/5
−1/5
1/5
−1/5
−3/103/10
+1
2/7
2/7−3/7
+31/2
+2+10
1/2
+2 +2+2+1 +2
+1+31/2
−1/2012
34
+1/2+3/2
+3/2+2 +5/24/7 7/2
+3/21/74/72/7
5/2+3/2
+2+1
−10
16/35
−18/351/35
1/3512/3518/354/35
3/2
3/2
+3/2
−3/2−1/21/2
2/5−2/5 7/2
7/2
4/3518/3512/351/35
−1/25/2
27/703/35
−5/14−6/35
−1/23/2
7/2
7/2−5/24/73/7
5/2−5/23/7
−4/7
−3/2−2
2/74/71/7
5/2−3/2
−1−2
18/35−1/35
−16/35
−3/21/5
−2/52/5
−3/2−1/2
3/2−3/2
7/2
1
−7/2
−1/22/5
−1/50
0−1−2
2/5
1/2−1/21/10
3/10−1/5
−2/5−3/2−1/21/2
5/2 3/2 1/2+1/22/5
1/5
−3/2−1/21/23/2
−1/10
−3/10
+1/22/5
2/5
+10
−1−2
0
+33
3+2
2+21+3/2
+3/2+1/2
+1/2 1/2−1/2−1/2+1/2+3/2
1/2 3 2
30
1/20
1/20
9/209/20
2 1
3−11/5
1/53/5
2
3
3
1
−3
−21/21/2
−3/2
2
1/2−1/2−3/2
−2
−11/2
−1/2−1/2−3/2
0
1−1
3/10
3/10−2/5
−3/2−1/2
00
1/41/4
−1/4−1/4
0
9/20
9/20
+1/2−1/2−3/2
−1/20−1/20
0
1/4
1/4−1/4
−1/4−3/2−1/2+1/2
1/2
−1/20
1
3/10
3/10
−3/2−1/2+1/2+3/2
+3/2+1/2−1/2−3/2
−2/5
+1+1+11/53/51/5
1/2
+3/2+1/2−1/2
+3/2
+3/2
−1/5
+1/26/355/14
−3/35
1/5
−3/7−1/21/23/2
5/22×3/2
2×2
3/2×3/2
−3
Figure 32.1: The sign convention is that of Wigner (Group Theory, Academic Press, New York, 1959), also used by Condon and Shortley (TheTheory of Atomic Spectra, Cambridge Univ. Press, New York, 1953), Rose (Elementary Theory of Angular Momentum, Wiley, New York, 1957),and Cohen (Tables of the Clebsch-Gordan Coefficients, North American Rockwell Science Center, Thousand Oaks, Calif., 1974). The coefficientshere have been calculated using computer programs written independently by Cohen and at LBNL.