Einführung in die Physik - Wagner, Zeilinger

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g





































ϕνσιζ

























































S

B



















=

·

F = ma



















F = mω 2 r

[F ] F ] =



11

   =

2

[F ] F ] = [

[v ] = [ v =1

]

−2

]



















kW

1

= 2.3884 10−14

·

m

10−1 t





















10−30 10−27 10−3 102 1024 1030 1042

10−23 10−7 102 109 1013 1017

→

10−15 10−6 101 1014 1025

I 2 107

·

T

1

= 6.022 1023

·



















x

n

xi ∆x

x¯ n

x¯

S =

S = x¯

n ¯ i=1 (x



−x) i

n

n





i=1

2(¯ 2(x¯

− x ) = 0 ⇐⇒ i

n

(¯x

i=1

2

− x ) = nx¯ i

 − i=1

xi = 0



















n n σ

2

−1

xw

1

≈ =

1 n

n

(x¯

i=1

n 2 x ¯ n

n

1 xi ) = n

  − −   −



2x¯

2

1 n

x¯

2

2xx x¯xi + xi

i=1

n

xi +

i=1

2

1 n

n

x2i = x¯2

i=1

 −   1 = n

2

− 2¯x

n

n

2

x ¯

1

2x¯

i=1

+ x2 = x2

2

− x¯

f ( f (x) =

√

σ π

exp

−

(x¯

− x)

2σ 2

2



σ

[x¯ 2σ

[¯x

± 2σ]

± σ]

3σ 3σ

[¯x

± 3σ]

n

x2i

xi +

i=1

σr = σ/¯ σ/ x¯

1

→ x¯ i=1



















R/ xi

σi

R(x1 , . . . , xn ) = A v = 1s1 t−1 n

σR

 ≤ |

σR R

Aji x ¯i σi

|

i=1

n

 ≤ || ii

i=1

σi x ¯i R

σR =

   ∂R ∂x 1

1 2 1

σ + ... +

      ∂R ∂x n

n

1

2

σn =



i=1

∂R ∂x i

2

σi2



n ¯ ji i i=1 x







































|v| = v

v (t + ∆t ∆t) a = lim ∆t→0 ∆t

− v(t) =

2 v r ˙ = v = 2 = ¨r t t

[a ] =

 2

a = at + an

at an





















a0 =

−g

r0 = r (t = 0) = 0 v0 = v (t = 0) = 0

a = z

  −  0 0 g

t=0

v =

  −  0 0 gt

r =

  −  0 0 1 gt 2 2

rn = nR1 rn = n2 R2





















rz (rx ) = rx tan(ϕ tan(ϕ) rz = tan(ϕ tan(ϕ) rx

−

−

1 2

v0−2 rx2 g cos2 (ϕ)

2rxg sin(ϕ sin(ϕ) = 2v02 cos2 (ϕ) cos(ϕ cos(ϕ)

−

rx g =0 v02 cos2 (ϕ)

=

⇒

rxmax

v02 sin(ϕ sin(ϕ)cos(ϕ )cos(ϕ) = g

H = rz (rxmax ) = v02 sin(2) ϕ/2 ϕ/2g W = 2rxmax = v02 sin(2ϕ sin(2ϕ) /g

xz

v0x





















v v =

s/ t =

(Rϕ) Rϕ)/ t = Rω

v = ω 

sin(ϑ) = Rω | ω × r| = | ω| |r| sin(ϑ

ϑ

r −2

×  ω

× r R

r

|v|

=











































=

t

a:t

t

a

F = ma

−

2















































 L

 N  = r L

× p

 = r N

× F 























M  = 1 R M

n



miri























∆ri



























































































































































































































































































































































































































































































































































 R F  R = µG F  N F N µG

 µG



















µH

µG

µr /R

R

 H  N F H = µH F N  R = µR F  N N N

[µR ] = [ ]

µH µH > µG



















 R F

mx¨ =

−kx

k x(t) = a 2 mω = k ω=

iωt

 ±

k/m

−maω ω

2 iωt

=

−ka

iωt



















ω (x) = a cos(ω cos(ω (t + t0 )) = a cos(ωt cos(ωt + ϕ) ϕ

T cos(ωt cos(ωt)) = cos(ω cos(ω(t + T )) T ))

2π

ωt + 2π 2π = ω(t + T ) T ) 2π T = ω

1 ω ν = = T 2π

[ν ] =

ν



=[

]

T = 0.1 2π ω = 2π



x1 = a cos(ω cos(ω1 t)





x1 + x2 = A cos(ω cos(ω1 t) + cos(ω cos(ω2 t) = 2a cos



(ω1

−ω )t 2

2

 

x2 = a cos(ω cos(ω2 t)

(ω1 + ω2 ) t cos 2





















N x=



N n=1 an

cos(ω cos(ωn t + ϕn )

ω x = a sin(ωt sin(ωt)) y = b sin(ωt sin(ωt + ϕ) sin(ωt sin(ωt + ϕ) = sin(ωt sin(ωt)cos( )cos(ϕ ϕ) cos(ωt cos(ωt)) sin( sin(ϕ ϕ) y/b = (x/a)cos( x/a)cos(ϕ ϕ) cos(ωt cos(ωt)) sin( sin(ϕ ϕ) sin(ϕ sin(ϕ) sin(ωt) ωt) sin( sin(ϕ ϕ) = (x/a) x/a) sin( sin(ϕ ϕ) y/b

−

2



2

2

 

sin (ϕ) sin (ωt) ωt) + cos (ωt) ωt )

=

x cos(ϕ cos(ϕ) a

x2 sin (ϕ) = 2 a x2 = 2 a 2

−

2 y 2 x + sin(ϕ sin(ϕ) b a 2xy y 2 x2 2 2 cos (ϕ) cos(ϕ cos(ϕ) + 2 + 2 sin (ϕ) ab b a 2 2xy y cos(ϕ cos(ϕ) + 2 ab b

− −

 



−

ϕ = 0 sin(ϕ) a∗ = a sin(ϕ



sin(ϕ sin(ϕ) ∗

b = b sin(ϕ sin(ϕ)

xy x2 y2 + a∗2 b∗2

cos(ϕ) = 1 − a2xyb cos(ϕ ∗ ∗





















ϕ = π/2 π/ 2 a=b 2

2

2

x +y =a f 1 = nf 2



 r  R + F  ma = F ¨ + r x˙ + kx = 0 mx x(t) = a iωt ω 2 + iωr/m i ωr/m + k/m = 0

−

ω0 = (k/m) k/m)1/2

 ± −

ω1,2 = ir/2 r/2m ω0 ω = ω0

r 2 /4m2



















γ = r/2 r/2m

ωd =

 −

ω02 γ 2 (x) = a −γt cos(ω cos(ωd t) T = 2π/ωd ae−1

γt = 1 τ

τ = 1/γ =

2m/r

τ

γ = r/2 r/2m

x(t) = a

α= −γt

(

ω = ir/2 r/2m x(t) = a −γt (1 + 1) 1) = 2a

αt

+

 − γ 2

−αt

ω02 x(t) = a

iiγt

ω1,2 = iγ iiα = a −γt

± iα αt

)

α = 0 −γt





















a=r

iϕ

ω r

k1

→ω

0

k2 k12 x























u(x, t) = u0 cos(ωt cos(ωt ω u x=0

− kx) kx)

u0 k

x u(t) = u0 cos(ωt cos(ωt))

x=0 T = 2π/ω ν = 1/T = ω/2 ω/ 2π t = t0

cos(k cos(k (x + λ)) = cos(kx cos(kx))

λ k(x + λ) = kx + 2π 1/λ = k/2 k/ 2π v

λ = 2π/k



















v = /(kv )/dk = v k = 2π/λ v

v (λ) = v

v (k) = v (λ)

v 2π v λ 2π v (2π/k (2π/k)) =v + =v + k k λ k k λ k 2 2π v 2π 2π v λ v = v + = v λ k λ k2 2π/λ λ 2π λ

= v

+k

= v

− v

+k v / k

−

=v v
v

∂ 2 u ∂ = ∂x 2 ∂x 2 ∂ u ∂ = ∂t 2 ∂t

− −

u0 ( k) sin( sin(ωt ωt



2

−

kx ) − kx)

=

u0 ω sin(ωt sin(ωt



−ω u

k2

k 2 /ω 2 = 1/v2

− kx) kx)

=

−k u 2

0

0

>v

cos(ωt cos(ωt

cos(ωt cos(ωt

kx ) − kx)

− kx) kx)

ω2 ∂ x2 u = (k 2 /ω 2 )∂ t2 u





















ν = vQ

1

−

ν 0 vG/v

ν = ν 0



vB 1+ v



vB





















xz





































F = ma

E K G

109

2

µ R

Einführung in die Physik - Wagner, Zeilinger

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