Formulario cuantica 2

Abraham Prado A01213521

∫|Ψ(x, t)|2 Φ(p, t) = 1√

2π~

∫∞−∞ e

−ipx~ Ψ(x, t)dx [f(x), p] = i~ dfdx L± ≡ Lx ± iLy

[Lx, Ly] = i~Lz [Ly, Lz] = i~Lx

[Lz, Lx] = i~Ly i~ ∂∂tΨ(ξ, t) =

(− ~

2m∇2 + V (ξ)

)Ψ(ξ, t) = EΨ(ξ, t)

ψn(x) =√

2L sin(nπxL );En = π2~2n2

2mL2 ψ0(x) =(mωπ~) 1

4 e−mωx2

2~ ;E0 = 12~ω [x, p] = i~

ax =√

mω2~(x+ i

mω px)

a†x =√

mω2~(x− i

mω px) [

ax, a†x

]= 1

〈f〉 =∫

Ψ(r, t)fΨ(r, t) = 〈Ψ|f |Ψ〉 j = − i~2m (Ψ∗∇Ψ−Ψ∇Ψ∗)

ρ(r, t) = Ψ∗(r, t)Ψ(r, t) p = ~k V (x) = 12mω

2x2 〈H〉 = 〈p2〉2m

〈x〉 =∫∞−∞ xρ(x)dx =

∫∞−∞ x |Ψ(x, t)|2 dx 〈T 〉 = − ~2

2m∫∞−∞Ψ∗ ∂

2Ψ(x,t)∂x2 dx

Ψ(x, t) =∑∞n=1 cnψ(x)e−

iEnt~

cn =∫ψn(x)∗f(x)dx cn =

√2a

∫ a0 sin(nπa x)Ψ(x, 0)dx 〈H〉 =

∑∞n=1 |cn|

2En =

∫Ψ(x, 0)∗HΨ(x, 0)dx

ψn(x) =(mωπ~) 1

4 1√2nn!Hn(ξ)e−

ξ22

|S(t)〉 = e−iE1t

~ |s1〉 |S(0)〉 =∑cn|sn〉

σ2P = 〈p2〉 − 〈p〉2 ci = 〈φi|ψ〉.∫x sin ax dx = sin ax

a2 − x cos axa

∫xn sin ax dx = −x

n

a cos ax+ na

∫xn−1 cos ax dx∫

x cos ax dx = cos axa2 + x sin ax

a + C∫xn cos ax dx = xn sin ax

a − na

∫xn−1 sin ax dx∫

sin b1x sin b2x dx = sin((b1−b2)x)2(b1−b2) − sin((b1+b2)x)

2(b1+b2)∫

sinn ax dx = − sinn−1 ax cos axna + n−1

n

∫sinn−2 ax dx∫

cosn ax dx = cosn−1 ax sin axna + n−1

n

∫cosn−2 ax dx

∫cos a1x cos a2x dx = sin(a1−a2)x

2(a1−a2) + sin(a1+a2)x2(a1+a2) + C

φ(x) =√mα~ e−mα|x|/~

2 ; E = −mα2

2~2

Φ(x, t) = 1√2π

∫∞−∞ φ(k)ei(kx−ωt)dk

ω =(

~k2

2m

)φ(k) = 1√

2π

∫∞−∞ e−ikxΨ(x, 0)dx

Φ(x, t) = 1√2π

∫∞−∞ ei(kx−

~k22m t)ψ(k)dk

pdav =√

2meqe |∆V |

λ = hmv

nλ = 2πr

hν = − me4

8ε20h

2

[ 1n2 − 1

m2

]K = mH2OCp∆T1−mH2OfrCp∆T2

∆T2

−Q = mCp∆T +Kcal∆T

∆T1 = Tmezcla − TH2Ocal

1

λ =√

2m(V0−E)~2

k =√

2mE~2

tan zz = tanh

√α2−z2

√α2−z2

si = −soMTds1ds0

= − −f2

(s0−f)2

2πλ ∆L.C.O. =

ψnlm = Rnl(r)Y ml (θ, φ)

Rnl(r) = −{(

2Zna0

)3 (n−l−1)!2n[(n+l)!]3

}(1/2)e−ρ/2ρlL2l+1

n+1 (ρ)

En = −Z2e2

2n2a0

〈rk〉 ≡∫∞

0 drr2+k[Rnl(r)]2

Y ml (θ, φ) = (−1)m√

2l+14π

(l−m)!(l+m)!P

ml (cos θ)eimφ

Pml (cos θ) = (1− cos2 θ)m/2 dm

d(cos θ)mPl(cos θ)

T = 1{1+[(k′2−k2)/4k2k′2] sin2 2k′a}

k =√

2mE~2

k′ =√

2m(E+V0)~2

T = 1{1+[(k2+κ2)2/4k2κ2] sinh2 2κa}

k =√

2m(−|E|+V0)~2

κ =√

2m|E|~2

T = 4kk′(k+k′)2

k =√

2mE~2

k′ =√

2m(E−V0)~2

T = exp

{−2∫ badx√

2m[V (x)−E]~2

}m d2

dx2 〈x〉 = −〈∇V (x)〉

exp(−iHt

~)

=∑K′ |K ′〉 exp

(−iEK′ t

~

)〈K ′|

|α, t0 = 0; t〉 = |a′〉 exp(−iEa′ t

~

)wa′′a′ = (Ea′′−Ea′ )

~

w = |e|Bmec

H|±〉 =(±~w

2)|±〉

〈x′|n〉 =(

1π1/4√

2nn!

)(1

xn+1/20

)(x′ − x2

0d

dx′)n exp

[− 1

2

(x′

x

)2]

x0 ≡√

~mω

L =√l(l + 1)~

S =√s(s+ 1)~

2

x′ = x−ut√1−u2/c2

t′ = t−(u/c2)x√1−u2/c2

v′x = vx−u1−vxu/c2 v′y = vy

√1−u/c2

1−vxu/c2

∆t′ = uL/c2√1−u2/c2

K = mc2√1−v2/c2

−mc2

p = mv√1−v2/c2

E =√

(pc)2 + (mc2)2

cn =( 2n+1

2) (n−m)!

(n+m)!∫ 1−1 f(z)Pmn (z)dz

Cn,m =∫ 2πϕ=0

∫ πθ=0 g(θ, ϕ)[Y mn (θ, ϕ)]∗ sin θdθdϕ

x = r√

2π3 (Y −1

1 − Y 11 ) y = ir

√2π3 (Y −1

1 + Y 11 ) z = r

√4π3 (Y 0

1 )

[Lx, Ly] = ~Lz[Li, Lj ] = ~εijkLk[L2, Li] = 0.

L± = Lx ± iLy [L2, L±] = 0

L±Ylm = ~√l(l + 1)−m(m± 1)Yl(m±1)

|L| =√l(l + 1)~

L2 = L+L− + L2z − ~Lz

[L+, L−] = 2~Lz

J+|j,m〉 =√j(j + 1)−m(m+ 1)~|j,m+ 1〉

J2 = J2x + J2

y + J2z

L2fmin = ~2[−l(−l − 1)]fmin

a = bmin(bmin − ~)

J+|a, bmax〉 = 0

[H,Lz] = 0

Lz = ~i

(x ∂∂y − y

∂∂x

)= ~

i∂∂ϕ

L± = ~e±iφ(± ∂∂θ + i cot(θ) ∂

∂φ

)Lx = L++L−

2

Ly = L+−L−2i

〈ψ|Lx|ψ〉

L+Yll = 0 [Li, r2] = [Li, p2] = [Li, r · p] = 0

L−Yl−l = 0 −l ≤ m ≤ l

Lzfml = ~mfml

L±fml = ~

√(l ∓m)(l ±m+ 1)fm±1

l

3

L2 = −~2[

1sin θ

∂∂θ

(sin θ ∂∂θ

)+ 1

sin2 θ∂2

∂φ2

][Li, rj ] = i~

∑k εijkLk

[Li, pj ] = i~∑k εijkpk∣∣∆k

k

∣∣� 1

4

Formulario cuantica 2

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