Paquetes de Onda

Ondas

A[x_] := x^2 /125 - Sin[35*x] /2+ 3*Cos[4*x]
Plot[ A[x], {x,0,2}, 
PlotRange -> {-4,4}
]

v= 0.5
t=1;
Plot[ {A[x],A[x - v*t]}, {x,0,2}, 
PlotRange -> {-4,4}, 
PlotStyle -> {RGBColor[0,0,0],RGBColor[1,0,0] }
]

Dibujo de Ondas Armonicas

v=2;
T= 2*π/v;
Onda[x_,t_] := Sin[x - v*t];

Manipulate[Plot[{Sin[x],Onda[x,t]}, {x,0,5 π}], 
{{t,0,"t"},0,T,Appearance->"Labeled"}]

Ejercicios:
1.  Hallar las velocidades  de avance de la onda (grupo o fase?)
  2.  Que pasa si en lugar de Sin[x]se pone Cos[x]?

Construccion de paquetes de ondas

Clear[Onda];
Onda[k_,ω_, x_,t_] := Sin[k*x - ω*t];

Δk = k/10 ;
Δω = ω/10;

k=1;
ω =2;
Plot[ {Onda[k,ω,x,0],Onda[k,ω,x,1] }, {x,0,4π}, 
PlotStyle -> {RGBColor[0,0,0],RGBColor[1,0,0] }
]

Δk = k/10 ;
Δω = ω/10;

Plot[ Onda[k,ω,x,1]+Onda[k+Δk,ω + Δω,x,1] , {x,0,45π} ]

Ejercicios
 1.  Hallar las velocidades  de avance de la onda (grupo o fase?)
 2.  Construir un paquete de ondas sumando muchas ondas
 3.   Avanza el paquete? A que velocidad?

Ondasum2[x_,t_] := Onda[k,ω,x,t]+Onda[k+Δk,ω + Δω,x,t]
Plot[ {Ondasum2[x,0],Ondasum2[x,1] }, {x,0,45π} ]

T= 4*π * Δk/Δω;
Manipulate[Plot[ {Ondasum2[x,0],Ondasum2[x,t]} ,
 {x,0,45 π} ],{{t,0,"t"},0,T,Δt/100,Appearance->"Labeled"}]

Paquete de Ondas

ClearAll[k,ω,x,t,wavepacket]

nondas = 10;
k=1;
ω=2;
Δk = k/nondas ;
Δω = ω/nondas;

δk=0.2;

Onda[k_,ω_, x_,t_] := Exp[I (k*x - ω*t)];

wavepacket[k_,ω_,x_,t_] := Sum[Exp[- (i Δk/δk)^2] Onda[k+i Δk,ω +i Δω,x,t], {i,-1 nondas, 1 nondas,1}]

Plot[Re[wavepacket[3,2,x,1]], {x,-35 π, 35 π}, PlotRange-> All]

Paquete de Ondas Gaussiano

G[x_] := (1/(Sqrt[σ(Sqrt[π])]))(E^(-((((x-a))^2)/(2(σ^2)))))

σ = 2;
a = 5;

Plot[  G[x] , {x,-3,15}]

Ejercicios
1.  Comprobar que (G^2) esta normalizado
3.  Que significan σ y a

Transformacion Fourier

Clear[x,k,Yx,Fk];
Yx[x_] = 1/(x^2 + 2);

Yk[k_] = 1/(Sqrt[2 π])  Integrate[ Yx[x] *Exp[I  k x], {x,-Infinity , Infinity}]

Plot[ Yx[x] , {x,-5,5}]
Plot[ Yk[k] , {k,-5,5}, PlotRange-> {0,1}]

Yfk[k_] = FourierTransform[Yx[x], x, k]

Plot[ {Yk[k],Yfk[k] }, {k,-5,5}]

Clear[x,k,σ,a, Gk];
G[x_] := (1/(Sqrt[σ(Sqrt[π])]))(E^(-((((x-a))^2)/(2(σ^2)))))

Gk[k_] = FourierTransform[G[x], x, k]

Clear[x,k,σ,a, Gk];
Gk[k_] := FourierTransform[G[x], x, k]  
σ=1 ;
a = 0 ;

Plot[ G[x] , {x,-3,3}]

Gk[1.5]

Plot[ Re[Gk[k]] , {k,-3,3}]

Ejercicios
1.  Comprobar que (Gk^2) esta normalizado
3.  Como varian los anchos y posiciones de la transformada?

Valores Medios

σ=1 ;
a = 2;

ClearAll[σ,a]

amed := Assuming[{σ>0,Element[σ,Reals]},Integrate[G[x]*x *G[x], {x,-Infinity, Infinity}]]

a2med := Assuming[{σ>0,Element[σ,Reals]},Integrate[G[x]*x^2*G[x], {x,-Infinity, Infinity}]]

amed

a2med

σx := a2med - (amed)^2

σx

P[x_] := -I h * G'[x]

P[2]

pmed := Assuming[{σ>0,Element[σ,Reals]},Integrate[G[x]*P[x], {x,-Infinity, Infinity}]]

pmed

P2[x_] := -I h * P'[x]

p2med := Assuming[{σ>0,Element[σ,Reals]},Integrate[G[x]*P2[x], {x,-Infinity, Infinity}]]

p2med

σp := p2med - (pmed)^2

σp

σx σp