B4ggg 4444 4444 556 PDF

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Summary

fghfgh gfgfh...

Description

W±

Z0

• • • •

ηµν =

(−1, 1, 1, 1)

n(θ, φ) jA

σ=

Z

dΩ

dσ , dΩ

dσ = dΩ

=

n jA dσ/dΩ

V (r) b

[θ, θ + dθ] [b, b + db] dΩ = sin θdθdφ

jA nd Ω = n sin θdθdφ = jA bdbdφ

b dσ = dΩ sin θ

   db     dθ 

( 0 V (r) = ∞

b(θ) = R sin α = R sin



r>R r≤R

π−θ 2



= −R cos θ/2

R2 dσ = 4 dΩ πR2

R

Z1 e Z2 e V (r) =

mr 2β˙ = mv0 b v0

r

∆p = 2mv0 sin

Z1 Z2 e2 4πǫ0 r

−→

dβ v0 b = 2 dt r

β

    Z ∞ θ θ Z1 Z2 e2 cos =− dt cos β ∇U = 2πǫ v b 2 2 0 0 −∞

    Z1 Z2 e2 d θ θ b= = cot cot 2 2 2 4πǫ0 mv02 b=0

dσ 1 d2 = 4 dΩ 16 sin (θ/2)

∽ 27.5 θ→0

θ

(H0 − E) |ψi = −V |ψi H0

V =0 H0 |φi = E |φi

|ψi = |φi +

V |ψi E − H0 ± iε

ε→0 1/(E − H0 ) Z

hx|

 ′  ′ 1 x x  V |ψi E − H0 ± iε Z  ′    1 x V (x′ ) x′ ψ = hx|φi + d 3 x′ hx| E − H0 ± iε

hx|ψi = hx|φi +

d 3 x′ hx|

G± (x, x′ ) = hx|

hx|pi = Z

 ′ 1 x E − H0 ± iε

1 eix·p/~ (2π~)3/2

  ′ 1 d 3 p hx|pi p x 2 E − p /2m ± iε Z 2m ′ 1 =− d 3 p ei(x−x )·p/~ 2 3 p − 2mE ± iε (2π~)

G± (x, x′ ) =

E

ε→0 r = |r| = |x − x′ | Z 2π Z 1 Z ∞ 2m 1 2 dp p dφ G± (x, x′ ) = − d(cos θ) eipr cos θ/~ 2 3 p − 2mE ± iε (2π~) 0 −1 0 Z ∞ ipr/~ 1 2mi pe dp = (2π~)3 r −∞ (p − ~k ∓ iε)(p + ~k ± iε) E = ~k k −1

O(ε2 )

f (z)

z = zn I

Γ

dz f (z) = 2πi

Γ

X

(f (z = zn ))

n

(f (z = zn ))

Γ p = ~k + iε ′

G± (x, x′ ) = −

1 2m hx|ψi = hx|φi − 4π ~2

Z

2m e±ik|x−x | ~2 4π|x − x′ | ′

   eik|x−x | V (x′ ) x′ ψ d x ′ |x − x | 3 ′

|x|V → 0

|x| → ∞ |x| ≫ |x′ |     ′ 1/2 ′ 2   x · x′ x − x′  = |x| 1 + |x | − 2 x · x +... ≈ |x| 1 − |x| |x|2 |x|2 |x − x′ |

hx|ψi = hx|φi −

1 2m eikr 4π ~2 r

r = |x|

Z

   ′ ′ d 3 x′ e−ik ·x V (x′ ) x′ ψ hx|ψi

hx|φi =

1 eik·x (2π)3/2

  eikr 1 ik·x ′ e + hx|ψi = f (k , k) (2π)3/2 r

f (k, k′ ) = −

1 2m   (2π)3 2 k′  V |ψi 4π ~

|x|

r dσ dσ = r 2

| dΩ = |f (k, k′ )|2 d Ω |

|j |j

dσ = |f (k, k′ )|2 dΩ

|ψi ≈ |φi hx|ψ i

f

(1)

(1)

1 2m eikr = hx|φi − 4π ~2 r

Z

   ′ ′ d 3 x′ e−ik ·x V (x′ ) x′ φ

1 2m   1 2m (k, k ) = − (2π)3 2 k′  V |ki = − 4π ~2 4π ~ ′

Z

′

′

d 3 x′ ei(k−k )·x V (x′ )

dσ/dΩ

V (x)

        hx|ψi(1) − hx|φi ≪ hx|ψi(1)  x′ = 0

|ψi ≈ |φi

( −V0 V (r) = 0

|V0 | ≪

~2 k ma

ka ≫ 1,

Z1 e

|x| ≤ a |x| > a

|V0 | ≪

~2 ma2

ka . 1

Z2

x′

ρ(r) Z1 e2 V (x ) = 4πǫ0 ′

Z

ρ(x′′ ) = Z1 ~cαEM d x |x′ − x′′ | 3 ′′

Z

d 3 x′′

ρ(x′′ ) |x′ − x′′ |

α

f

(1)

Z

(∆k) = AZ1 ~cα

3 ′

d x

Z

d 3 x′′ ei(∆k)·x

A = −m/(2π~2 ) q

∆k = k − k′

1 f (q) = Z |

Z

3

ρ(x′′ ) |x′ − x′′ | r = x − x′

iq·x × AZ

d x ρ(x)e {z

′

}

Z

1 Z2 ~cα

|

d3 r

{z

e−iq·r r }

ρ(x) = Z2 δ 3 (x)

F

1 (q) = Z

dσ = |F dΩ

Z

(q)|2

d 3 x ρ(x)eiq·x



dσ dΩ

ρ(x) 

ρ(x)

q V (x)

(−∂ρ ∂ ρ + µ2 )φ = 0,

µ=

mc ~

φ φ = φ(r, t)

(∇2 − µ2 )φ = 0 φ= c1

−→

1 ∂2 (rφ) = µ2 φ r ∂r 2

c1 −µr c2 µr e + e r r

c2

c2 = 0

V (r) =

g 2 e−µr 4π r

g

f

(1)

2m 1 (q) = − 2 ~ q

Z

0

= ~/(me a0 c)

∞

dr rV (r) sin(qr) = −

g 2 2m 1 4π ~2 q 2 + µ2

g4 dσ (1) = (4π)2 dΩ



2m ~2

2

1 [2k 2 (1 − cos θ) + µ2 ]2

q = |q| = 2k sin θ/2 µ→0 µ−1 m = ~µ/c

P · P 6= −(E/c)2 + p2

m G=

g g

−1 P · P + m2 c 2

ik·x

e

=

∞ X

iℓ (2ℓ + 1)Jℓ (kr)Pℓ (cos θ)

ℓ=0

Jℓ (kr)

Pℓ (cos θ)

P0 = 1,

P1 = cos θ,

1 P2 = (3 cos2 θ − 1) 2

d 2 uℓ ℓ(ℓ + 1) 2m − uℓ + 2 [E − V (r )]uℓ = 0, ~ dr 2 r2

uℓ (r ) = rJℓ (kr ) V (r)

Vr → 0

r→∞

 ( sin(kr)  sin kr − ℓ π2 kr = cos( lim Jℓ (kr) = kr ) r→∞ kr kr

hx|ψi =

∞ X

ℓ ℓ

cℓ Jℓ (kr)Pℓ (cos θ)

ℓ=0

cℓ ∞ 1X (2ℓ + 1)Pℓ (cos θ)eiδℓ sin δℓ k ℓ=0 4π X (2ℓ + 1) sin2 δℓ σ= 2 k

f (θ) =

ℓ

δℓ

ℓ

nπ δℓ σℓ ≤

4π (2ℓ + 1) k2



 n + 21 π

n

s

p = ~k b

p · x = ~kb

ℓ=0 ka ∽ kr ≪ 1

ℓ=0

s a

V (r) = V0 δ(r − a) s − u0 ≡ u

ε→0

~2 d 2 u + V0 δ(r − a)u = Eu 2m dr 2 a+ε 2mV0 du  u(a) = lim  ~2 dr ε→0 a−ε

r =a−ε

r = a+ε

u1 = A sin(kr + δ0 ) u2 = sin(kr) δ0

u1

u2 r=a

A sin(ka + δ0 ) = sin(ka) 2mV0 sin(ka) Ak cos(ka + δ0 ) − k cos(ka) = ~2

cot(ka + δ0 ) − cot(ka) = s

2mV0 1 ~2 k

ka ≪ 1 δ0 = −

cot x ≈ 1/x − x/3 kaΦ , 1+Φ

Φ=

2mV0 a ~2

H |ψi i ≡ |ii

|ψf i ≡ |f i

Mif = hi| H |f i Γ=

2π dN | hi| H |f i |2 dEf ~

Ef

Mif

−g

g αEM =

e2 1 ≈ 137 4πǫ0 ~c g

gEM =

p √ 4παEM = e/ ǫ0 ~c Q

1/29

αS ≈ 1

Qg αW ≈

Mif G=

m

−1 , P · P + m2 c 2

γ W ±, Z 0 g

•

P · P 6= −(E/c)2 + p2

g

Q gW gS

e+ e− → e+ e−

e+ e− → e+ e−

•

e+ e− → e+ e−

e+ e− → e+ e− •

e+ e− → e+ e−

e+ e− → e+ e−

•

e+ e− → µ+ µ−

e+ e− → µ+ µ− •

e+ e− →

e+ e− →

•

e− e− → e− e−

e− e− → e− e−

•

e− e− → e− e− + µ+ µ−

e− e− → e− e− + µ+ µ−

•

γ → e+ e−

γ → e+ e−

•

γγ → γγ

γγ → γγ

•

•

• µ−

µ− • τ−

τ− τ− µ

−

e+ e− → µ+ µ−

P1 = (Ee /c, p),

P2 = (Ee /c, −p),

Pγ = P1 + P2 = (2Ee /c, 0) c2 /4E e2

m=0

Mif = ~3 c ×

= g2

×

~3 c 3 4E e2

j = 2ve = 2c2 pe /Ee dσ =

Ee 2π dN |Mif |2 2c2 pe ~ dEf

dN dpµ dN dpµ 4π pµ2 = = 3 dpµ dEf dEf dEf (2π~) Eµ ≫ mµ c 2 E 2µ = pµ2 c2 + mµ2 c4

Eµ ≈ 12 Ef Eµ dEµ = c2 pµ dpµ

−→

1 E 1 dpµ dpµ = 2 µ = 2 dEµ dEf 2c pµ

σ = π~2 c2 α

α2 , s

s = (2E)2

= g 2 /4π

s

∆t ∽ Γ−1 ∆E ∆t ∽ ~

−→

∆E ∽ ~Γ Γ~/c2

Γ

Γ~/c2

  δℓ ≈ n + 12 π

σ E0

 d cot δℓ  (E − E0 ) + . . . cot δℓ (E) ≈ cot δℓ (E0 ) + | {z } dE E=E0 =0

 dδℓ  2 = dE E=E0 Γ

cot δℓ (E) ≈ −

 1 dδℓ  2 (E − E0 ) + · · · ≈ − (E − E0 ) Γ sin δℓ2 dE E=E0 | {z } =1

σℓ =

Γ2 /4 4π 4π 4π 1 2 (2ℓ + 1) ≈ (2ℓ + 1) sin δ = (2ℓ + 1) ℓ k2 1 + cot2 δℓ (E − E0 )2 + Γ2 /4 k2 k2

σ=

Γi Γf /4 2j + 1 4π k 2 (2s1 + 1)(2s2 + 1) (E − E0 )2 + Γ2 /4

• Γi • Γf • Γ=

P

j

Γj = τ −1

Γj

• E • E0 • j

s1

s2

• k E Γ

Bj =

Γj Γ

E0

• • •

ρ(r) = a

ρ0 −( + e r−a)/d

1

d d

989.6

/c

mp = 983.3

2

/c2

mn =

A Z

Z A N

A =Z +N

a r = r0 A1/3 ,

r0 ≈ 1.2

A •

r V ∝r

3

A1/3 A

• •

∽ 2r0 π0, π±

∽ ∽

/c2

/c2 M (A, Z)c2 = Zmp c2 + (A − Z)mn c2 − B(A, Z) B(A, Z ) = aV A − aS A2/3 − aA |{z} | {z } |

aV , as , aA , aC , δ

Z2 (N − Z)2 − aC 1/3 ± δ(N, Z) A } | {z } {zA } | {z

• E = −c1 n + 4πR2 T R ∝ n1/3

T

n n∽A

•

N =Z •

•

Z 2 /r

nn

r ∽ A1/3

pp N Z

+δ

0

−δ

aV

aS

A

aA

aC

Z

A & 30 A . 30

δ√ 11.2/ A

A N =Z

(N −Z) ≪ A

n ¯i =

1 eβ(E−µ) − 1 mp ≈ mn ≡ m

E=

g(E)dE =

~2 k 2 p2 = 2m 2m

(2s + 1)V m3/2 1/2 √ E dE 2π 2 ~3

EF

N n ¯i

E ≪ EF N =

X

n ¯i =

n

Z

EF

dE g(E) = 0

EF =

hEi =

X



n ¯ i Ei =

6π 2 n 2s + 1 Z

0

i

EF

(2s + 1)V m3/2 2 3/2 E √ 3 F 2π 2 ~3

2/3

~2 2m

3 dE g(E)E = N EF 5

N = Z = A/2 EF =

~2 2m



3π 2 N V

2/3

=

~2 2mr 20



 9π 2/3 8

V = (4π/3)r 3 = (4π/3)r 30 A N ≈Z hEi = hEN i + hEZ i =

N 5/3 + Z 5/3 3 EF 5 (A/2)2/3

δ = (Z − N )/A N = A(1  − δ)/2  (1 ± x)5/3 ≈ 1 ± 35 x + 2!1 53 53 − 1 x2 E

Z = A(1 + δ)/2

(N − Z)2 1 = EF 3 A EF /3 ≈ 11.2

δ

N Z

N Z N Z Z =N A B(A, Z) Z A   ∂B 2(A − 2Z )(−2) 2Z − aC 1/3 = 0 = −aA A ∂Z A A aC 2/3 N =1+ A Z 2aA NZ 208

N =Z

Q/c2 =

X i

Mi (A, Z) −

X

Mf (A, Z)

f

Q>0

N dN = −ΓN dt Γ N (t) = N0 e−Γt τ [t, t + dt] P (t)dt = −

1 dN dt = Γe−Γt dt N0 dt R∞

τ = hti = R0∞ 0

N = (A, B, C)T

dt tP (t) 1 = Γ dt P (t) A B

C

dN = RN dt R b

eRt = U et R U −1

N = N(0)eRt , b R

Γ

R

U

R

N m ≫ Γ~/c2

4 2

α

A−4 −→Z−2

A Z

+24

Q   2 2 ~ ∂ − + V (r ) hx|ψi = Q hx|ψi 2m ∂r 2 exp η(r ) exp η Q=−

 ~2  ′′ η + (η ′ )2 + V (r ) 2m r

V (r)

α V (r) =

(

r < Ra r > Ra

Z1 Z2 ~cαr

η η ′′ ≪ (η ′ )2

r

P = G G=

r

2m ~2

Z

Rb

dr

Ra

Ra

p

|hRb |ψi|2 = e−2G |hRa|ψ i|2

V (r) − Q,

Ra ≈ r0 A1/3 ,

Rb =

Z1 Z2 ~cα Q

Rb Q≪V G≈ Q≪1

α

r

2m p Z1 Z2 ~cα ~2

Z1 = 2

Z

Rb

Ra

dr r −1/2 ≈

√ Z1 Z2 e2 2m √ 2πǫ0 Q Ra/Rb ∽

Z2 ≡ Z Z Γ = Γ0 e−2G BZ log Γ = A − √ Q

Z

A Q→0

B

Q→∞ V ℓ(ℓ + 1)/(2mr 2 )

β−

N A Z

A + e− + ν¯e −→ Z+1

A Z

A −→ Z−1

A Z

+ e− −→ A Z−1

β+

+ e+ + νe

+ νe

β+ Q

2me c2

β−

Q>0

Q>0 A A

A 2δ

β− • GF ≈ 1.17 × 10−5

−2

•

• hx|φe i ≡ φe = eipe ·x/~ ,

hx|φν i ≡ φν = eipν ·x/~

• Ee + Eν = Q

|Ψi i = |ψi i ,

|Ψf i = |ψf i × |φe i × |φν i

Mif = hΨf | H |Ψi i = GF

Z

d 3 x φe∗ φ∗ν ψf∗ψi p·x ≪ 1

Mif ≈ GF

Z

d 3 x ψf∗ψi ≡ GF M

M |M |j −j ′ | ≤ 1

d2 N =

d 3 pe d 3 pν p2e p2ν dpe dpν = (2π~)3 (2π~)3 4π 4 ~6

|=1

dN =

1 pe2 dpe 3 4 6 4π ~ c

Z

dEν Eν2δ(Ee + Eν − Q) =

p2e (Q − Ee )2 dpe dEν 4π 4 ~6 c3

dEf = dQ = dEν

Γ(pe )dpe =

|2 GF2 |M F (ZD , pe )pe2 (Q − Ee )2 dpe 3 7 3 2π ~ c

F (ZD , pe )

Γ ∝ Q5

∽ α

β γ

γ

A Z

•

A Z

•

−→A−4 Z−2

A + e− + ν¯e −→ Z+1 A Z

• A Z

• A

+42

A −→ Z−1

+ e+ + νe

A + e− −→ Z−1

+ νe

Z A

A = 4n A = 4n + 1 A = 4n + 2 A = 4n + 3

232 237 238 235

1.41 × 1010 2.14 × 106 4.47 × 109 7.04 × 108 237

A

pp p + p −→ D + e+ + νe

≈

E E

~cαEM ≈ 0.7 2 ≈ 4×109 K

/2

t1/2 ≈ 12 2 1

Q

+31

−→4

+n ≈ 108 K

E Z/2

=

=

~cαEM (Z/2)2 2r0 (A/2)1/3

A/2 A ≈ 200

A ∽ N

N δ

N

N

238

∽ γ 238

∗

+ n −→238

−→238

+γ

235

235 92

236 92 235

+ n −→236

∗

 84% −→

16% 236

−→



+γ

235

238

235

235

+ n −→236

∗

−→140

+93

+ 3n + γ

235

NZ 235 − 92 N ≈ ≈ 1.55 Z 92 140

93 140 58

A 93 41

235

∆E = M (235, 92) − M (140, 58) − M (93, 41) − 2mn ≈ 200 c2 235

200

235

238 238

m M

u

U = um/M

θ

m v =u+U v v2 = v2

k

+ v2

⊥

= (u cos θ + U )2 + (u sin θ)2 = u2 + 2U u cos θ + U 2 v 2 /v 2 θ

m

hcos θi = −

Z

1

d(cos θ) cos θ = 0

−1

m 

E E



=

m2 + M 2 u2 + U 2 = 2 (m + M )2 (u + U ) m ≡ mn M

2

2

A=1 log2 (

/0.1

) ≈ 23

λ

2mD mD ≈ 1865 ¯0 c¯ u D D+ c d¯

•

2

D0

/c D− c¯d D

q q¯ •

π+ π0 c¯ c

m < 2mD

π−

n=2 αS ≈ 0.97

n=1

J/ψ c¯ c

/c2

mc = 1870 mb = 5280

/c

α q

αS

q

q

αS ≈ 1

αS α

≈ 1/137

τ ∽ 10−22

•

τ ∽ 10−18

•

τ ∽ 10−10

•

K + −→ π + + π 0

[∆S = −1]

Σ− −→ n + π

[∆S = +1]

u¯ s

dds

ud

udd

u¯ u,dd¯ −

ud

GF Mif

αS > αW > α ∽ 10−6

Υ b¯b

n=2 n=1 αS ≈ 0.57

2

• • • • 21st

u, d, c, s, t, b ℓ

α γ ×8 g

±

W Z0

∞ ∽ 10−15 ∽ 10−18 ∞

∽1

∽ 10−2 ∽1 ∽ 10−8 ∽ 10−41

Q

X X X

X X X

P

X X X

X X X

X X X

X X X

C

CP

X X X

X CP

e−

µ−

νe νµ

∽ 200 2.2 µ µ− −→ e− + ν¯e + νµ

µ− τ− ∽ 3500 2.9 × 10−13

ντ e−ν¯e

τ

τ − −→

  π − + ντ    π − + π 0 + ν −

+

τ −

 π + π + π + ντ    K − + ν τ

µ−ν¯µ

τ−

W Z µ−

τ−

e− νe

Γ ∝ Q5 ∝ m5CM

Γ(τ − → e− + ν¯e + ντ ) = Γ(µ− → e− + ν¯e + νµ )



mτ mµ

5 gµ



gτ gµ

2

=

gτ

−1   5 τµ τµ mτ B(τ − → e− + ν¯e + ντ ) Γ(τ − ) = 0.178 ≈ 0.999 − − − mµ B(µ → e + ν¯e + νµ ) Γ(µ ) ττ ττ | {z } =0.178

Γ = τ −1 ∝ g 2 m5

Le = N (e− ) + N (νe ) − N (e+ ) − N (¯ νe )

Lµ = N (µ− ) + N (νµ ) − N (µ+ ) − N (¯ νµ ) Lτ = N (τ − ) + N (ντ ) − N (τ + ) − N (¯ ντ ) Lℓ = Le + Lµ + Lτ

α

= 1/137

  1 √ (|r¯ r i + |g¯ g i + bb¯ ) 3 αS ∽ 1

q q¯

e+ e−

∽

W± Z0

γ

Q

Gγ = −

1 , Q2

GW = −

Q m(W ± , Z 0 )c2 ≫ Q Gγ ≫ GW Q

Pe = (Ee /c, pe ),

1 Q2 + m2 c2 W±

Z0

Gγ /GW

P′e = (E ′e /c, pe′ ),

Pq = xPp 0...