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8/15/2019 Dispense di Meccanica Quantistica
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• x1, x2, x3 r,θ,φ d3x = dx1dx2dx3 = r
2 sin θdrdθdφ
• a,b,c A,B,C α, β, γ Γ, ∆, Θ • a,b, c A,B,C α,β Γ,∆ • a · b a × b• ∇ = ( ∂ ∂x1 , ∂ ∂x2 , ∂ ∂x3 ) φ ∇φ v
∇ · v v ∇× v• a, b, c A,B,C
• N N0
Z
Q
R
C
•
|ψ ψ| ψ|φ
•
1 = 10−8 1 = 1.6 · 10−13
•
= 1.054589(6) · 10−27erg · s•
e = 1.602189(5) · 10−19C = 5.34426 · 10−9 • me = 9.10953(5) · 10−27g = 0.511003(1) /c2
m p = 938.280(3) /c2
mn = 939.573(3) /c2
• c = 2.99792458(1) · 1010
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L3
L2
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Bext Bint Bext Bint
Γ(z )
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C
L2
L2
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M
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N q m
q
m = 1.76 · 108 C
g
−2 · 10−19C < q < −1019C
10−27
−e e = 1.602189(5) · 10−19C
me = 9.10953(5) · 10−27g e
1 = 1.602189(5)
·10−13
106
mec2 = 0.511003(1) MeV
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+
m p = 1.67265(1) · 10−24g , m pc2 = 938.280(3) MeV 10−8cm =1 Å Z A
42 +2e
4m p
α
42
++
10−13cm = 1 fm
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R
R = σT 4 , σ = 5.67 · 10−5 ergcm2 · s · K4
I (λ) λ
λmax = α
T , α = 0.29cm · K
I (λ)
I (λ) λ
I (λ) ∝ T λ4
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λ λ
ν ν + dν
ρ(ν )dν = ̄N (ν )dν
̄ ν N (ν )dν L
(cos θ1, cos θ2, cos θ3) L
nλ
2 = L
ni = 2L
λ cos θi , i = 1, 2, 3
n21 + n22 + n
23 =
4L2
λ2 =
4ν 2
c2 L2
R = 2νLc
ν ν + dν R R + dR dR = 2Ldν
c
1
84πR2dR =
1
84π
2Lν
c
22Ldν
c =
4πν 2
c3 L3dν
N (ν )dν = 8πν 2dν
c3
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ρ(ν ) ε̄
f (ε) = Ce− εkT
k = 1.38 · 10−16
ε ε + dε
ε̄ =
´ ∞0
εf (ε)dε´ ∞0
f (ε)dε
β = 1/kT
ε̄ =´ ∞0 εe−βεdε´ ∞
0 e−βεdε
= − ddβ
logˆ ∞
0
e−βεdε = ddβ
log β = 1β
= kT
ρ(ν )dν = 8πkT ν 2
c3dν
λ = c/ν
dν = − c
λ2 dλ
ρ(λ)dλ = −ρ(ν )dν
ρ(λ) = 8πkT
λ4
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ε = hν h
εn = nε = nhν
ε̄ =
∞n=0
εne−βεn
∞n=0
e−βεn= − d
dβ log
∞n=0
e−βεn
= − ddβ
log∞n=0
(e−βhν )n = − ddβ
log 1
1 − e−βhν
∞
n=0
xn = 1
1 − x
ε̄ = hν
ehνkT − 1 =
⇒ ρ(ν )dν = 8πν 2
c3hν
ehνkT − 1dν
ρ(ν ) ρ(λ)
ρ(λ) = 8πhc
λ51
e hckTλ − 1
h
h = 6.62618(4) · 10−27erg · s
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=
h
2π = 1.054589(6) · 10−27
erg · s
ρ ≡ˆ ∞
0
ρ(ν )dν = 8πh
c3
ˆ ∞0
ν 3dν
ehνkT − 1
a = 8πhc3 b = h/kT
ρ = aˆ ∞
0
ν 3dν
ebν − 1 = a ˆ
∞
0
ν 3e−bν
1 − e−bν dν
= a
ˆ ∞0
ν 3e−bν ∞n=0
(e−bν )n = a∞n=1
ˆ ∞0
ν 3e−bνndν
= −a∞n=1
d3
d(bn)3
ˆ ∞0
e−bnν dν = −a∞n=1
d3
d(bn)31
bn
= 6a
b4
∞n=1
1
n4
∞n=1
1
n4 =
π4
90
ρ = 8π5hk4
15c3 T 4
T 4
h,k,c σ
α = 8πhc γ = hc/kT
ρ(λ) = α
λ5(eγ/λ − 1)−1
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xex 5(ex − 1)
dρ
dλ = − α
λ6(eγ/λ − 1)−1
5 −
γ λ
eγ/λ
eγ/λ − 1
ρ(λ) λ = ∞ λ = λmax
5(ex − 1) = xex
x = γ/λ
x0 = 4.96511...
λmaxT = hc
kx0= cost.
λmax = cost.
T
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V
εc = 1
2mev
2 = eV
ν ≥ ν 0 ν 0
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ν 0
εc ∝ (ν − ν 0) I
N N ∝ I ν ≥ ν 0
10−9
I I 0
εc ∝ (I − I 0)
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E = mc2 ε = hν
c
Ν0Ν
Εc
ε =
p2c2 + m2c4 m = 0
ε = pc
ε = hν , p = hν
c =
h
λ
ε0
hν < ε0
hν ≥ ε0 ≡ hν 0
εc = h(ν − ν 0)
N ∝ I
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λ < λ λ − λ θ
λ − λ = λc(1 − cos θ) λc λc = 2.4·10−10 θ
γ e−
E γ + E e = E γ + E
e
pγ + pe = pγ + p
e
E γ = hν ν E e =mec
2
hν + mec2 = hν + Mc2
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ν n = cR
1
4 − 1
n2
, n = 3, 4, 5...
R R =
109677.576cm−1
ν n = cR
1 − 1
n2
, n = 2, 3, 4,...
ν n = cR
1
9 − 1
n2
, n = 4, 5, 6... ,
ν n = cR 1
16 − 1
n2
, n = 5, 6, 7... ,
ν n = cR
1
25 − 1
n2
, n = 6, 7, 8... ,
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ν m,n = cR 1
m2 − 1
n2 , m = 1, 2, 3, ... n = m + 1.m + 2,...
E 1, E 2, E 3,...
E m E n hν = |E m − E n|
l = p
×r
|l| = n
n = 1, 2, 3,... n +e −e
E n = T n + V n = 1
2mev
2n −
e2
rn
e2
r2n=
mev2n
rn=⇒ mev2n =
e2
rn
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E n = −12
e2
rn= −1
2mev
2n
vn =
mern
rn = 2
mee2n2
n2 1/n
vn = e2
1
n
v1 ≈ 2.2 · 108
r1 ≈ 0.53
E n = −mee4
2 21
n2
n m
ν m,n = E n − E m
h =
πmee4
3
1
m2 − 1
n2
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E 1 = −13.59
E ion = E ∞ − E 1 = 13.59eV
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E
ν
E = hν = ω
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ω = 2πν
p = |p| = hλ
= 2π
λ
k = p
k = |k| =2π/λ
e±i(k·x−ωt) = e±i
(p·x−Et)
p = k = 2π
λ
k
k
E = pc
ν = c
λ =⇒ ω = c|k|
ν = E
λp
E = 1
2mv2 =
p2
2m
ω = k2
2m
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r 2πr
2πr = nλ
λ = 2π /p l = pr
l = n
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•
•
•
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ψ(x, t) ∈ C •
•
P ∝ |ψ|2
x t
P (x, t)d3x ∝ |ψ(x, t)|2d3x d3x =dx1dx2dx3 x
• ˆ
R3P (x, t)d3x = 1
ˆ R3
|ψ(x, t)|2d3x = C < ∞
ψ̌(x, t) = 1√
C ψ(x, t)
P (x, t) = |ψ̌(x, t)|2
ψ̌ C |ψ|2 P
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•
ψ eiα α |ψ̌|2
|ψ|2
ψ
ψ
|ψ|2 |ψ|2 < M
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C
ψ1
p1
ψ2
p2 N = |α|2 + |β |2 ψ ψ1,2 ψ p1 |α|2/N p2 |β |2/N
m
p = mv v x
λ = h/p
kx − ωt k = 2π/λ kx − ωt
A sin(kx − ωt) B cos(kx − ωt)Ce
i(kx−ωt)
De−i(kx
−ωt)
A,B,C,D
p
∂ 2ψ
∂t2 = γ
∂ 2ψ
∂x2
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γ
ω2
= γk2
=⇒ γ = p2
4m2
γ
ω k
ω = k2
2m
∂ψ
∂t = γ
∂ 2ψ
∂x2
γ Cei(kx−ωt)
−iω = −γk2 =⇒ γ = i 2m
γ
i ∂ψ
∂t = −
2
2m
∂ 2ψ
∂x2
F = ma
γ = − i2m
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ψ = Cei(kx−ωt)
ωψ = 2k2
2m ψ
Eψ = p2
2mψ
E → i ∂ ∂t
p → −i ∂ ∂x
ψ(x, t) = Cei(k·x−ωt)
ω = |k|2
2m
i ∂ψ
∂t = −
2
2m∇2ψ
∇2 = ∂ 2∂x2 + ∂ 2
∂y2 + ∂ 2
∂z2
E
→ i
∂
∂tp → −i ∇ ∇ ( ∂ ∂x ,
∂ ∂y ,
∂ ∂z
)
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vf = ω
k =
p
2m
vg = dω
dk =
p
m
|ψ(x, t)|2 = |C |2
ˆ |ψ(x, t)|2d3x
´ Ω1
|ψ(x, t)|2d3x´ Ω2
|ψ(x, t)|2d3x Ω1 Ω2 Ω1 Ω2
p
x
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p ± ∆p p − ∆p p + ∆p
∆p
k
C (k)
ψ(x, t) =
ˆ C (k)ei(k·x−ωt)d3k
ψ(x, t) =
ˆ k0+∆kk0−∆k
C (k)ei(kx−ω(k)t)dk
ω k ω(k) = k2
2m
k [k0 − ∆k, k0 + ∆k] C (k)
C = C (k0) ξ = ∆k = k − k0
ω(k) = ω0 + ω1ξ + ...
ω0 = ω(k0) = k202m
, ω1 = dω
dk
k0
= k0
m
ψ(x, t) = Cei(k0x−ω0t)ˆ ∆k−∆k
ei(x−ω1t)ξdξ
ψ(x, t) = 2C sin[(x − ω1t)∆k]
x − ω1t ei(k0x−ω0t)
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A(x, 0)
ei(k0x−ω0t)
t = 0
A(x, 0) = 2C sin(x∆k)
x
x = 0 2C ∆k
xn = nπ
∆k , n = ±1, ±2,...
[−π/∆k,π/∆k]
∆x = 2π
∆k
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|ψ(x, t)|2 = |A(x, t)|2
|ei(k0x−ω0t)|2 = 1 A(x, t) x
ω1 = dω
dk
k0
= p0m
(−∞, +∞) ˆ
+∞
−∞|ψ(x, t)|2dx = 4|C |2∆k
ˆ +∞
−∞sin2 η
η2 dη
η = (x−ω1t)∆k π
ψnorm(x, t) = 1√
π∆k
sin(x − ω1t)∆kx − ω1t e
i(k0x−ω0t)
∆x ≈ 2π/∆k
∆ p = ∆k
∆x∆ p ≈ h
∆x ∆k
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ψ F
V F = −∇V
E = p2
2m + V
E → E = i ∂ ∂t
p → p = −i ∇x → x = x
x
xψ = xψ
i ∂ψ
∂t = −
2
2m∇2ψ + V (x)ψ
N q = (q 1, ....q N ) L(q, q̇ ) p = ( p1,...,pN )
pi = ∂ L∂ q̇ i
H ( p, q ) = p · q̇ − L
E = H
i ∂ψ
∂t = Hψ
H H ( p, q ) pi → pi = −i ∂ ∂qi q qi = q i
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V ψ∗
i ψ∗ ∂
∂tψ = −
2
2mψ∗∇2ψ + V (x)ψ∗ψ
ψ
−i ψ ∂ ∂t
ψ∗ = − 2
2mψ∇2ψ∗ + V (x)ψ∗ψ
i ∂
∂t(ψ∗ψ) = −
2
2m(ψ∗∇2ψ − ψ∇2ψ∗)
= − 2
2m∇ · (ψ∗∇ψ − ψ∇ψ∗)
P = |ψ|2 = ψ∗ψ J = − i 2m(ψ∗∇ψ − ψ∇ψ∗) = (ψ∗vψ) v v ≡ p/m
∂ P ∂t
+ ∇ ·J = 0
P J
V
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Ω ⊂ R3
Ω Ω
Ω [0, L] L
L → ∞
ψ(x + L, t) = ψ(x, t) , i = 1, 2, 3
p = k
ψ = Cei(kx−ωt)
ei[k(x+L)+ωt] = ei(kx−ωt) =⇒ eiLk = 1 = e2πni
k = kn = 2π
L n , n ∈ Z
k
p = pn = 2π
L n ω = ωn =
k2n2m
= 2 π2
mL2n2
ψn(x, t) = Cei(knx−ωnt) = C e
2πL i(xn− πmLn2t)
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[0, L]
ˆ L0
|ψn(x, t)|2dx = |C |2ˆ L
0
dx = |C |2L
C = L−1/2
ψn(x, t) = 1√
Lei(
2πL xn−ωnt)
x
P (x, t) = 1L
L2(0, L)
[0, L]
ˆ L0
ψ∗n(x, t)ψn(x, t)dx = ei(ωn−ωn )t
L
ˆ L0
e2πL i(n−n)xdx
=
n = n ei(ωn−ωn )t2πi(n−n)
e2πL
(n−n)xL
0= 0
n = n 1L ´
L
0 dx = 1
ˆ L0
ψ∗n(x, t)ψn(x, t)dx = δ n,n
δ n,n
+∞n=−∞
ψn(x, t)ψ∗n(x
, t) = 1
L
+∞n=−∞
e2πL in(x−x) = δ (x − x)
ψ
∈ L2(0, L)
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[0, L]
ψ(x, t) =n
anψn(x, t)
ψ [0, L] ψ
{ψn}
an = (ψn, ψ) =
ˆ L0
ψ∗n(x, t)ψ(x, t)dx
ψ an p = 2π L n ψ
|an
|2
ψ
ˆ L0
|ψ|2dx = 1
n
|an|2 = 1
p = −i ∂/∂x
−i ∂ ∂x
ψn(x, t) = −i √ L
∂ ∂x
ei( 2πL xn−ωnt) = 2π L
n 1√ L
ei( 2πL xn−ωnt)
pψn(x, t) = pnψn(x, t)
p
ψn pn
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N q = (q 1,...,q N )
L(q, q̇, t)
p = ( p1,...,pN )
pi = ∂ L∂q i
H ( p, q, t) = p · q̇ − L(q, q̇, t)
˙ pi = ∂H
∂q i, q̇ i = −∂H
∂pi
f ( p, q, t) g( p, q, t)
{f, g}P =i
∂f
∂q i
∂g
∂pi− ∂f
∂pi
∂g
∂q i
= −{g, f }P
{q i, p j}P = δ i,j
df
dt = {H, f }P + ∂ f
∂t
∂H ∂t = 0
dH dt
= 0 {H, H }P = 0 F
dF dt
= {H, F }P
dF dt
= 0
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H
t |ψ(t) ∈ H α ∈ C |ψ α|ψ
t
|ψ1 |ψ2 α|ψ1 + β |ψ2 α, β
|ψ̌ = |ψψ ψ < ∞
ψ = ∞ H
F ( p, q, t) F
H
F
H
F |ψ f k F
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F F
|ψ
F
F|k = f k|k
f k
• P k = |k|ψ|2 • P k =
gki=1 |ki|ψ|2 gk
f k {|ki} f k
• ρ(k) = |k|ψ|2 F f (k) f (k) + dk ρ(k)dk =
|k
|ψ
|2dk
k|ψ
F f k |k
|ψ f k
f k |k
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|ψ(t)
i ddt
|ψ(t) = H|ψ(t) H
H = H (p, q)
q p
qi p j
[q, p] = 0
H =N i=1
p2i2m
+ V (q)
q p
qp pq
1
2{q, p} = qp + pq
2
{·, ·} p q qp
H
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{|i}
F
F ij = j|F|i i, j F
F k,k = k|F|k F |k |ψ = F|k
|k
|ψ =k k|ψ|k
k| |ψ |k
F |ψ |k |ψ F |k F k,k |k |k F |ψ
P k→k = |k|
F̂ |k|
2
k|F̂ † F̂ |k = |F k,k
|2
lk|F̂ †|ll|F̂ |k = |F k,k
|2
l |F k,l|2 F |k |k F k,k F
F |ψ1 |ψ2
|ψ1 |ψ2 F
P ψ1→ψ2 = |ψ2
|F
|ψ1
|2
ψ1|F2|ψ1
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x = (x1, x2, x3)
x
x|x = x|x
|x x|x = δ (x − x)
ˆ R3
|xx|d3x = 1
|ψ ψ(x) = x|ψ
|ψ
ˆ R3
|ψ(x)|2d3x = 1
ψ1|ψ2 = ˆ R3
ψ1|xx|ψ2d3x = ˆ R3
ψ∗1(x)ψ2(x)d3x
x|x
x|x = ψx(x) = δ (x − x)
L2
p =
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−i ∇
F
F H F(x)
ψ(x) = x|ψ |ψ φ(x) = x|φ |φ = F|ψ
F(x)ψ(x) = φ(x)
x F |ψ x
φ(x) = x|φ = x|F|ψ = ˆ x|F|xx|ψd3x=
ˆ x|F|xψ(x)d3x
F(x)ψ(x) =
ˆ x|F|xψ(x)d3x
F(x)
ψ(x) R3
F(x) ψ(x) ψ(x) �