F2004 formulas final_v4
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Transcript of F2004 formulas final_v4
ss —Draft— THX 1138
AbrahamPrado1213521 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
AbrahamPrado1213521 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
]∫ ∞−∞
e−x2+bx+c dx =
√π eb
2/4+c A =
cos θ cosψ cosφ sinψ + sinφ sin θ cosψ sinφ sinψ − cosφ sin θ cosψ− cos θ sinψ cosφ cosψ − sinφ sin θ sinψ sinφ cosψ + cosφ sin θ sinψ
sin θ − sinφ cos θ cosφ cos θ
D(α, β, γ) =(e−i
(α+γ)2 cos β2 −e−i
(α−γ)2 sin β
2ei
(α−γ)2 sin β
2 ei(α+γ)
2 cos β2
)
Rnl = 1rρl+1e−ρν(ρ), ν(ρ) =
∞∑j=0
cjρj , ρ = r
na, cj+1 = 2(j + l + 1− n)
(j + 1)(j + 2l + 2)cj ,
sin(α± β) = sinα cosβ ± cosα sin β cos(α± β) = cosα cosβ ∓ sinα cosβ
tan(α± β) = tanα± tan β1∓ tanα tan β cosh ix = 1
2 (eix + e−ix) = cosx sinh ix = 12 (eix − e−ix) = i sin x∫
tanh ax dx = a−1 ln(cosh ax)∫
coth ax dx = a−1 ln(sinh ax) ex = cosh x+ sinh x
e−x = cosh x− sinh x.∫
sin a1x cos a2x dx = −cos((a1 − a2)x)2(a1 − a2) − cos((a1 + a2)x)
2(a1 + a2) 〈p〉 = md〈x〉dt
〈p〉 = ~i
∫ ∞−∞
dxψ∗(x)∂xψ(x) ~ = 1,054× 10−34J · s kg ·m2
s~ = 6,582× 10−15eV · s me = 9,10938 · 10−31kg
dk
dxkL(α)n (x) = (−1)kL(α+k)
n−k (x) H2n(x) = (−1)n 22n n! L(−1/2)n (x2)
Ln(x) = ex
n!dn
dxn(e−xxn
)Hn(x) = (−1)nex
2 dn
dxne−x
2Hn+1(x) = 2xHn(x)−H ′n(x) L(α)
n (x) = x−αex
n!dn
dxn(e−xxn+α)
A−1 == 1ad− bc
(d −b−c a
)A−1 =
(ek − fh) (ch− bk) (bf − ce)(fg − dk) (ak − cg) (cd− af)(dh− eg) (gb− ah) (ae− bd)
U†U = 1 A =
(A+A†
2
)+(A−A†
2
)H† = H HT = H∗ AT = ±A D = TAT−1 En = ~2n(n+ 1)
ma2 ..
2