Bc0103 Aula 07 Função de Onda Interpretação Probabilistica e o Principio Da Incerteza

7/24/2019 Bc0103 Aula 07 Função de Onda Interpretação Probabilistica e o Principio Da Incerteza

http://slidepdf.com/reader/full/bc0103-aula-07-funcao-de-onda-interpretacao-probabilistica-e-o-principio 1/39



•

•

Ψ (x, t)

Ψ

Ψ

•

Ψ

• m p

λ = h

p

Ψ (x, t) = A ei(kx−wt) = A eik(x−vt)

A v = wk



Ψ

Ψ A

A cos (kx − wt) .

v = w/k = wλ/2π

x

1 2 3 4 5 6 x

1.0

0.5

0.5

1.0

y0

Ψ



k A (k)

Ψ

Ψ −→

y (x, t)

E (x, t) Ψ

•

vg = dw

dk

E = hν = w

p = h

λ=

h

2π λ2π

= k



E = p2

2m=

2k2

2m= w

⇒ w = k2

2m

vg = d

dk

k2

2m

=

km

= pm

vfase = λν = w

k=

k

2m=

p

2m=

v

2

vg vfase w = kvfase

vg = dw

dk= vfase + k

dvfase

dk

w = E/ = k2

2m

vfase = wk =

k2m = v2

kdvfase

dk=

k

2m=

v

2

vg = p/m = v



dvfase

dk= 0 =⇒ vg = vfase

⇒ vg = vfase

K = p2

2m, p = mv ,

v c

E =

p2c2 + m2c4 , p = mv

1 − v2

c2

w = w (k)

p =

k

w = E

=

1

p2c2 + m2c4

= 1

2k2c2 + m2c4



vg = dw

dk=

2c2k

E =

c2 p

E

p = mv

1 − v2

c2

p2(1 − v2

c2) = m2v2

v2

m2 +

p2

c2

= p2

⇒ v2 = c4 p2

p2c2 + m2c4 =

c4 p2

E 2 = v2

g



•



I = cρ

= c0 | −→E |2



P (x) ∼| E |2

•

•



• P 1 P 2

P 1 + P 2

x

P 12 = P 1 + P 2

• Ψ (x, t)

P (x)

≡|Ψ

|2= Ψ∗Ψ

Ψ∗Ψ



P (x) dx =| Ψ |2dx

x x + dx

P 1 = | Ψ1 |2

P 2 = | Ψ2 |2

P 12 =| Ψ1 + Ψ2 |2

= P 1 + P 2

P (x) =| Ψ |2

Ψ (x , y , t)

P (x , y , t) =| Ψ (x , y , t) |2

t



P (x , y , t)

•

∆x



•

−→

k

k p

p = k = h

2π

2π

λ=

h

λ

∆x → ∆k

Ψ (x, t) = A ei(kx−wt)

k

∆x −→ ∆k



∆x ∆k

∆x

∼

1

∆k

∆x ∆k = 2π

x

∆x x

∆x

k

A (k)



Ψ (x, t) =

ˆ +∞

−∞A (k) e

i(kx−wt)dk

A (k)

k

∆x

∆x ∆k

A (k)

A (k) = 1

2√ πδ

k

−δ

≤k

≤k + δ

0 k < k − δ, ou k > k + δ

| A (k) |2 k



t = 0

Ψ (x, 0) =

ˆ +∞

−∞A (k) e

ikxdk =

ˆ k+δ

k−δ

1

2√

πδe

ikxdk

= e

ikx sin δx

√ πδx

P (x, 0) =| Ψ (x, 0) |2= sin2 δx

πδx2

∆x ≈

π

δ=⇒ ∆x ∆k ≈ 2π

∆k = 2δ

p = k



∆x ∆ p ≈ 2π

∆

∆k

1λ = k

2π

∆x

∆k

x = 0 t = 0

x = 0

∆x x = 0

t = 0

∆x

∆x P (x) =| Ψ |2

∆ p

k

∆ p = ∆k

k

| A (k) |2

A (k) k | A (k) |2

k p = k | A (k) |2

f ( p) f ( p) = 2π | A

p

|2

p



x δ → ∞ ∆x → 0, ∆ p → ∞

k δ → 0 ∆x → ∞, ∆ p → 0

∆x ∆ p

∆x ∆ p ≥

2

•

w

f (x, t) = eikx−iwt

x = 0



f (0, t) ≡ f (t) = e−iwt

f (t) =

ˆ +∞

−∞A (w) e−iwt

dw

A (w)

A (w) =

1 w − δw ≤ w ≤ w + δw

0 w < w − δw, ou w > w + δw

|f (t)

|2

|A (w)

|2

∆w ∆t ≈ 2π

E = w

∆E ∆t ≈ 2π .



∆x ∆

∆x ∆k ∼ 1

∆w ∆t ∼ 1

∆x

1∆k ∆w

1∆

∆x ∆k ≥ 1

2

∆w ∆t ≥ 12

∆x ∆ px

≥

2

∆E ∆t ≥

2



y z

∆y ∆ py ≥

2

∆z ∆ pz ≥

2

Ψ (x, t) = A ei(kx−wt) = A ei2π(xλ

−νt)

∆x

px = hλ = kx ∆ px = ∆kx

∆x = ∞ ⇒ ∆ px = 0

∆E = h∆ν

∆t

∆E = 0 ⇒ ∆t = ∞



•

↓↑

∆x ∆ p ≥

2





pγ = h

λ

2θ

⇒ px

− pγ sinθ

≤ px

≤ pγ sinθ

⇒∆ px = 2 pγ sinθ = 2

h

λsinθ

λ → ∆ px

∆ px



x

∆x

∆x sinθ = λ

⇒ ∆x = λsinθ

∆ px∆x = 2h

λ

sinθ × λ

sinθ

= 2h >

2 ∆x λ ∆ px

⇒ ∆x ∆ px

• ∆x ∆ p

P (x) f ( p)

P (x) =| Ψ (x) |2

f ( p)

Ψ (x) =ˆ +

∞−∞

A (k) eikxdk =ˆ +

∞−∞

1

A

p

e i p xdp

f ( p) = 2π

| A

p

|2 .



∆x ∆ p

∆x ∆ p =

2

∆x = σx =

(x − x)2

=

x2 − (x)2

∆ p = σ p =

p2

− ( p)2

.

σx,p



v = 103m/s 0.1%

me = 9.1 × 10−31kg

∆ pe = me∆v =

9.1 × 10−31

kg×10−

3

×10

3

m/s = 9.1×10−31

kg

∆x ≥

2∆ p=

1.05 × 10−34J · s

1.82 × 10−30kg · m/s= 0.58 × 10−4m = 0.058 mm

106

m = 1 g v = 103m/s

0.1%

∆ p = m∆v = 10−3kg× 10−3 × 103m/s = 10−3kg · m

s

∆x ≥

2∆ p=

1.05 × 10−34J · s

2 × 10−3kg · m/s= 0.55 × 10−31m

L

∆ p = 0



∆x = L

∆ p ≥

2L

(∆ p)2 ≥ p2 − ( p)2

p = 0

E min > (∆ p)2

2m

=

2

8m L2

= 1

(2π)

2

h2

8m L2

n2

λn = L =⇒ pn = h/λn = n h/2L



E n = p2

n

2m=

n2h2

8m L2, n = 1, 2, ...

E min > 1

(2π)2 E 1

n = 1

• L ∼ 1A

E min >

2

2me L2 ≈ 3.8 eV

E 1 = h2

8me L2 =

(6.63)2 × 10−68J 2 · s2

8 × 9.1 × 10−31kg × 10−20m2

= 0.6 × 10−17J ≈ 38 eV

•



∆x → 0 ∆ p → ∞

∆x = a

(∆ p)2 = p2 − p2



∆x (∆ p)

(∆ p)2 ≈ p2

∆x ∆ p ≥

2

(∆ p)2 ≥ 2

4a2

p2

p2 ≈

2

4a2

E = p2

2m− e2

4π0r

E =

2

8m a2 − e

2

4π0a

dE

da= −

2

4m a3 +

e2

4π0a2 = 0

⇒ a = 14

4π0

2

m e2

= a0

4 ≈ 0.13

o

A

a0 = 0.533o

A

r > a

2a0 ≈ 1o

A



E > E = 2m

2

e2

4π0a2

2

− 4m

2

e2

4π0

2

= −2m

2 e2

4π0

2

= −54.4 eV

E 0 = −13.6 E 0 > E

∆x ∆ p ≈ a = a0 E = E 0

•

∆E ∆t ≥

2

∆w ∆t ≥ 1

2

∆t

∆w ∆ν



∆t

∆ν

∆E

∆t

∆t≥

2∆E

τ ∼ 10−8s

∆t =

∆E ≥

2∆t= 1.05 × 10−34J · s

2 × 10−8s= 0.525 × 10−26J

≥ 0.33 × 10−7eV

∆ν = ∆E

h≥ 0.525 × 10−26J

6, 63 × 10−34J · s= 8 × 106s−1



3s

589.0 589.

3 p 3 p32

3 p12



∆ν

∆ν ν

= 8 × 106s−1

cλ

= 8 × 106s−1 × 589 × 10−9

3 × 108 = 1.6 × 10−8

589.0

E 3 p − E 3s ≈ hν = ∆E

∆ν ν

= 0.33 × 10−7eV

1.6 × 10−8 = 2.1 eV

∼ ∆E E

∆E ≥

2τ

τ

τ ∼

2∆E



W

e+e− → W +W − → qqqq ou qqlν

• ⇒

∆y



y

∆y py

∆y sinθ = λ

∆ py ≈ 2 py = 2 p sinθ = 2 p λ

∆y

p = hλ

∆ py = 2h

∆y=⇒ ∆ py∆y = 2h >

2

∆ py



Ψ (x) k A (k)

Ψ (x) =ˆ +

∞−∞ A (k) eikx

dk

P (x) =| Ψ (x) |2 =

ˆ +∞−∞

A∗

k

e−ikx

dk ˆ +∞

−∞A (k) e

ikxdk

ˆ +∞

−∞| Ψ (x) |2 dx =

ˆ +∞

−∞dkA∗

kˆ +∞

−∞dk A (k)

ˆ +∞

−∞e−i

k−k

xdx

=

ˆ +∞−∞

dk ˆ +∞

−∞dk A

∗ k

A (k) 2πδ

k − k

=

ˆ +∞−∞

dk 2π | A (k) |2=

ˆ +∞−∞

dp2π

| A

p

|2

δ

k − k

= 12πˆ +

∞−∞ e−ik

−kx

dx

ˆ +∞−∞

dkA∗ k A (k) δ

k − k

= A∗ (k) A (k)

ˆ +∞−∞

P (x) dx =

ˆ +∞−∞

dp f ( p)

P (x) =| Ψ (x) |2

2π

p

2

Documents

Bc0103 Aula 07 Função de Onda Interpretação Probabilistica e o Principio Da Incerteza