Dispense di Meccanica Quantistica

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) �

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Dispense di Meccanica Quantistica