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Guia de estadistica

Función de probabilidad de 1 partícula

Defino la función de Maxwell-Boltzmann

f1=Function[{m,T,v},4π*(m/(2π*1.38064852×10^-23*T))^(3/2)v^2E^(-m*v^2/(2T*1.38064852×10^-23))]

Compruebo que esté correctamente normalizada

Integrate[f1[16*1.66054*10^(-27),100,v],{v,0,∞}]

Obtengo la probabilidad de tener una particula con velocidad mayor a 1000 m/s para las distintas temperaturas

T100=NIntegrate[f1[16*1.66054*10^(-27),100,v],{v,1000,∞}]

T1000=NIntegrate[f1[16*1.66054*10^(-27),1000,v],{v,1000,∞}]

T10000=NIntegrate[f1[16*1.66054*10^(-27),10000,v],{v,1000,∞}]

Obtengo la cantidad de moleculas promedio que tendrán más de 1000 m/s de velocidad

NA=6.02*10^(23);

nprod={NA*T100,NA*T1000,NA*T10000}

Representación gráfica

Para distintas temperaturas, grafico la función de probabilidad

Plot[{f1[16*1.66054*10^(-27),100,v],f1[16*1.66054*10^(-27),1000,v],f1[16*1.66054*10^(-27),10000,v]},{v,0,5000},PlotRange->All,PlotLabel-> "Oxigeno",AxesLabel-> {"v(m/s)","f(s/m)"},PlotLegends->{"T = 100K","T = 1000K","T = 10000K"},Filling->Axis ]

Guia de cuántica

Comparación entre Wien, Planck y Rayleigh-Jeans

Defino las tres densidades de energía

Wien=Function[{ν,h,c,kB,T},(8π*h*ν^3)/c^3*E^(-h*ν/(kB*T))]

Planck=Function[{ν,h,c,kB,T},(8π*h*ν^3)/c^3*1/(E^(h*ν/(kB*T))-1)]

RayleighJeans=Function[{ν,h,c,kB,T},(8π*h*ν^3)/c^3*kB*T/(h*ν)]

Veamos la ley de Desplazamiento de Wien

Manipulate[Plot[{10^(12)Wien[ν*10^(12),6.63*10^(-34),3*10^8,1.4*10^(-23),T]},{ν,0,1400},AxesLabel->{"ν [THz]","(Subscript[u, ν]) [\!\(\*FractionBox[\(J\), \(\*SuperscriptBox[\(m\), \(3\)]\\ THz\)]\)]"},PlotRange->{{0,1400},{0,0.002}},PlotLegends->{"Wien"}],{T,3000,6000}]

Veamos que pasa a una temperatura fija

Regimen clásico

Plot[{10^(6)Wien[ν*10^(6),6.63*10^(-34),3*10^8,1.4*10^(-23),5800],10^(6)Planck[ν*10^(6),6.63*10^(-34),3*10^8,1.4*10^(-23),5800],10^(6)RayleighJeans[ν*10^(6),6.63*10^(-34),3*10^8,1.4*10^(-23),5800]},{ν,0,100},AxesLabel->{"ν [MHz]","(Subscript[u, ν]) [\!\(\*FractionBox[\(J\), \(\*SuperscriptBox[\(m\), \(3\)]\\ MHz\)]\)]"}(*,PlotRange->{{0,1400},{0,0.002}}*),PlotLegends->{"Wien","Planck","Rayleigh-Jeans"},PlotLabel->"T=5800K"]

Catastrofe ultravioleta

Plot[{10^(12)Wien[ν*10^(12),6.63*10^(-34),3*10^8,1.4*10^(-23),5800],10^(12)Planck[ν*10^(12),6.63*10^(-34),3*10^8,1.4*10^(-23),5800],10^(12)RayleighJeans[ν*10^(12),6.63*10^(-34),3*10^8,1.4*10^(-23),5800]},{ν,0,1400},AxesLabel->{"ν [THz]","(Subscript[u, ν]) [\!\(\*FractionBox[\(J\), \(\*SuperscriptBox[\(m\), \(3\)]\\ THz\)]\)]"},PlotRange->{{0,1400},{0,0.002}},PlotLegends->{"Wien","Planck","Rayleigh-Jeans"},PlotLabel->"T=5800K"]

Comparo utilizando el parametro x=hv/kT

Manipulate[Plot[{10^(12)Planck[ν*10^(12),6.63*10^(-34),3*10^8,1.4*10^(-23),(6.63*10^(-34)*ν*10^(12))/(1.4*10^(-23)*x)],10^(12)Wien[ν*10^(12),6.63*10^(-34),3*10^8,1.4*10^(-23),(6.63*10^(-34)*ν*10^(12))/(1.4*10^(-23)*x)],10^(12)RayleighJeans[ν*10^(12),6.63*10^(-34),3*10^8,1.4*10^(-23),(6.63*10^(-34)*ν*10^(12))/(1.4*10^(-23)*x)]},{ν,0,1400},AxesLabel->{"ν [THz]","(Subscript[u, ν]) [\!\(\*FractionBox[\(J\), \(\*SuperscriptBox[\(m\), \(3\)]\\ THz\)]\)]"},PlotLegends->{"Planck","Wien","Rayleigh-Jeans"},PlotLabel->"Comparación hv/kT fijo "],{x,0.01,5}]