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.