ss —Draft— THX 1138

Sj = ~2σj σiσj = δij + i

∑k

εijkσk σ = [σx, σy, σz] σ ·A =[

Az Ax − iAyAx + iAy −Az

]σy =

[0 −ii 0

]σz =

[1 00 −1

]σx =

[0 11 0

]

(σ ·A)(σ ·B) = (A ·B)I + iσ · (A×B) χ = aχ+ + bχ− χ+ =[10

]χ− =

[01

]u = 1√

2

(11

)v = 1√

2

(1−1

)χ =

(a+ b√

2

)u+

(a− b√

2

)v [σx, σy] = 2iεijkσk

{σx, σy} = 2δijσ0 σiσi = 1 Sn ={±~

2

}Rz =

cosϕ − sinϕ 0sinϕ cosϕ 0

0 0 1

[Si, Sj ] = i~εijkSk S2|s,m〉 = ~2|s,m〉 Sz|s,m〉 = ~m|s,m〉 S± = Sx ± iSy

S±|s,m〉 = ~√s(s+ 1)−m(m± 1)|s,m± 1〉 Sxχ± = ~

2χ∓[

a√1− a2eiϕ

]eiα Sy = ~

2

(0 −ii 0

)e±iθ = cos θ ± i sin θ S = [Sx, Sy, Sz]

Sn = Sx sin θ cosϕ+ Sy sin θ sinϕ+ Sz cos θ χn+ =[

cos θ2eiϕ sin θ

2

]χn− =

[e−iϕ sin θ

2− cos θ2

]~Sn = ~

2

[cos θ e−iϕ sin θ

eiϕ sin θ − cos θ

]

f(~r − δ~r) = f(~r)[1− i

~δ~r · ~p] Ry(ε) =

1− ε2

2 0 −ε0 1 0−ε 0 1− ε2

2

Sy = 12i (S+ − S−) S±χ∓ = ~χ± S±χ± = 0 Sxχ± = ~

2χz Syχ± = ∓ ~2iχ∓ Szχ = ±~

2χ±

S2χ± = 34~

2χ± χ†χ = 1 〈Sx〉 = χ†Sxχ 〈S2j 〉 = ~2

4 bj =√

(s+ j)(s+ 1− j) S+ = ~

0 bs 0 · · · 00 0 bs−1 · · · 0...

......

. . ....

0 0 0 · · · b−s+10 0 0 · · · 0

〈Sx〉 = ~Re(ab∗) 〈Sy〉 = −~Im(ab∗)

〈S2〉 = ~2s(s+ 1) S2 = 34~

2(

1 00 1

)↑=∣∣∣∣12 1

2

⟩ ∫ ∞−∞

xne−αx2dx = 1 · 3 · 5 · (n+ 1)π1/2

2n/2α(n+1)/2 , n = 2k∫ ∞

0xne−ax

2dx =

12 Γ(n+1

2)/a

n+12 (n > −1, a > 0)

(2k−1)!!2k+1ak

√πa (n = 2k, a > 0)

k!2ak+1 (n = 2k + 1 , a > 0)

Rn(β) = exp{− i~βn · L

}Rn(β) =

[cos β2 − inz sin β

2 −(inx + ny) sin β2

−(inx − ny) sin β2 cos β2 + inz sin β

2

]T~a = exp

{− i~~a · ~p

}a =

(cos θe−iϕ/2

sin θ2eiϕ/2

)Rz =

(eiα/2 0

0 e−iα/2

)

Rr = cos(α

2

)I + i sin

(α2

)n · σ S− = ~

0 0 · · · 0 0bs 0 · · · 0 00 bs−1 · · · 0 0...

.... . .

......

0 0 · · · b−s+1 0

Sz = ~

s 0 · · · 00 s− 1 · · · 0...

.... . .

...0 0 · · · −s

|s1 m1〉|s2 m2〉 =

∑s

Cs1s2sm1m2m|sm〉 |sm〉 =

∑m1+m2=m

Cs1s2sm1m2m|s1 m1〉|s2 m2〉 (σk)ml = 〈sm|Sk|s l〉

s~

c+ =(χ

(k)+

)†χ χ

(y)+ = 1√

2

(1i

)χ

(y)− = 1√

2

(1−i

)〈Lx〉 = 0 〈Ly〉 = 0 φ(p) = 1

(2π~)3/2

∫e−i(p·r)/~ψ(r)dr3

P = |U〉〈U |eiω1t + |V 〉〈V |eiω2t Pn(x) = 12nn!

dn

dxn[(x2 − 1)n

]θ = cos−1

{ml√l(l + 1)

}ω = γB0 S = ~

2 [sinα cos γB0t − sinα sin γB0t cosα]

χ(t) = Aχ+e−iE+t

~ +Bχ−e−iE−t

~ H = −γ~S · ~B ~µ = γ~S H = −γB0~2

[1 00 −1

]|χ(t)〉 =

[cos α2 e

iγB0t2

sin α2 e

−iγB0t2

]X = 〈X|σx|X〉 ~τ = ~µ× ~B U = −~µ · ~B γ = q

2me

HY m` (θ, ϕ) = ~2

2I `(`+ 1)Y m` (θ, ϕ) ∆f = 1r2

∂

∂r

(r2 ∂f

∂r

)+ 1r2 sinϕ

∂

∂ϕ

(sinϕ∂f

∂ϕ

)+ 1r2 sin2 ϕ

∂2f

∂θ2 R(α, β, γ)

00r

=

r cosα sin βr sinα sin βr cosβ

,

...

1

ss —Draft— THX 1138

P−m` = (−1)m (`−m)!(`+m)!P

m`

Y 00 (θ, ϕ) = 1

2

√1π

Y −11 (θ, ϕ) = 1

2

√3

2π sin θ e−iϕ Y 01 (θ, ϕ) = 1

2

√3π

cos θ Y 11 (θ, ϕ) = −1

2

√3

2π sin θ eiϕ

Y −22 (θ, ϕ) = 1

4

√152π sin2 θ e−2iϕ Y −1

2 (θ, ϕ) = 12

√152π sin θ cos θ e−iϕ Y 0

2 (θ, ϕ) = 14

√5π

(3 cos2 θ − 1) Y 12 (θ, ϕ) = −1

2

√152π sin θ cos θ eiϕ

Y 22 (θ, ϕ) = 1

4

√152π sin2 θ e2iϕ eX =

∞∑k=0

1k!X

k a† =

0 0 0 0 · · ·√1 0 0 0 · · ·

0√

2 0 0 · · ·0 0

√3 0 · · ·

......

......

. . .

a =

0√

1 0 0 · · ·0 0

√2 0 · · ·

0 0 0√

3 · · ·0 0 0 0 · · ·...

......

.... . .

H = ~ω(a†a+ 1

2

)x =

√~

2mω(a† + a

)

p = i

√~

2mω(a† − a

)a|n〉 =

√n|n− 1〉 a†|n〉 =

√n+ 1|n+ 1〉 |n〉 =

(a†)n

√n!|0〉 Hn(ξ) = (−1)neξ

2 dn

dξn e−ξ2

H|n〉 = (n+ 12)~ω|n〉

Pm` (x) = (−1)m

2``! (1− x2)m/2 d`+m

dx`+m(x2 − 1)`. P−m` (x) = (−1)m (`−m)!

(`+m)!Pm` (x).

P 00 (cos θ) = 1 P 0

1 (cos θ) = cos θ P 11 (cos θ) = − sin θ

P 02 (cos θ) = 1

2 (3 cos2 θ − 1) P 12 (cos θ) = −3 cos θ sin θ P 2

2 (cos θ) = 3 sin2 θ

P (r) = [Rn`(r)]2r2 |e3〉 = |v3〉 − |e1〉〈e1|v3〉 − |e2〉〈e2|v3〉||v3〉 − |e1〉〈e1|v3〉 − |e2〉〈e2|v3〉|

µ± = 12

[a+ d±

√(a− d)2 + 4bc

]|v±〉 =

[(a− d∓

√(a− d)2 + 4bc

)/2c

1

]...

2