SetDirectory[ NotebookDirectory[] ]

Exemplo 2.2

Integrate[(x(c-x))^2,{x,0,c},Assumptions->c>0]

exp=Integrate[Sin[n Pi x/c]x(c-x),{x,0,c},Assumptions->{c>0,n∈Integers}]

FullSimplify[exp,Assumptions->n∈Integers]

FullSimplify[exp/.n->2i,Assumptions->i∈Integers]

FullSimplify[exp/.n->2i+1,Assumptions->i∈Integers]

a=10

Plot[Sqrt[30/a^5]x(a-x),{x,0,10}]

Plot[Sqrt[2/a]Sin[Pi x/a],{x,0,10}]

Plot[{Sqrt[30/a^5]x(a-x),Sqrt[2/a]Sin[Pi x/a]},{x,0,10},PlotStyle->{Blue,Red}]

Plot[{(Sqrt[30/a^5]x(a-x))^2,(Sqrt[2/a]Sin[Pi x/a])^2},{x,0,10},PlotStyle->{Blue,Red}]

Sum[1/n^4,{n,1,Infinity,2}]

Sum[1/n^6,{n,1,Infinity,2}]

Problema 2.4

2/c Integrate[x Sin[n Pi x/c]^2,{x,0,c},Assumptions->{c>0,n∈Integers}]

Integrate[y^2 Sin[y]^2,{y,0,n Pi},Assumptions->{n∈Integers}]

Integrate[y^2 Sin[y]^2,{y,0,n Pi}]

Problema 2.7

Integrate[Piecewise[{{x,x<a/2},{(a-x),a/2≤x<a}},0]Sin[n Pi x/a],{x,0,a},Assumptions->{a>0,n∈Integers}]

Integrate[x Sin[n Pi x/a],{x,0,a/2},Assumptions->{a>0,n∈Integers}]

f[x_,a_,m_]:=Total[ Table[(-1)^((n-1)/2)4Sqrt[6]/(n Pi)^2Sqrt[2/a]Sin[n Pi x/a],{n,1,m,2}]]

Plot[f[x,10,41],{x,0,10}]

16 6/Pi^4//N

Sum[1/n^2,{n,1,Infinity,2}]

Problema 2.8, ocupando a metade direita (não deve mudar nada)

Sum[ (4(Cos[n Pi]-Cos[n Pi/2])^2 )/(n^2 Pi^2),{n,1,Infinity}]

Table[ 2(Cos[n Pi]-Cos[n Pi/2])/(n Pi),{n,1,12}]

Problema 2.29

mc2=1000 (* nucleon, em MeV *)

hbarc=2 10^(-3) (* em MeV A *)

a=10^-5 (* em A *)

v0=100 (* em MeV *)

sol1=Plot[{Sqrt[-(e+v0)/e],-Tan[Sqrt[2 mc2] a/hbarc Sqrt[e+v0]]},{e,-v0,0},PlotRange->{-1,6},PlotStyle->{Red,Black},PlotLabels->{"raiz","tangente"}]

Export["pocoSol1.jpg",sol1]

FindRoot[Sqrt[-(e+v0)/e]==-Tan[Sqrt[2 mc2] a/hbarc Sqrt[e+v0]],{e,-18}]

z0=a Sqrt[2mc2 v0]/hbarc

sol2=Plot[{Cot[z],-Sqrt[(z0/z)^2-1]},{z,0,z0},PlotStyle->{Black,Red},PlotLabels->{"tangente","raiz"}]

Export["pocoSol2.jpg",sol2]

erg[z_]:=-v0+z^2hbarc^2/(2mc2 a^2)

erg[2.02]

raiz=FindRoot[Cot[z]==-Sqrt[(z0/z)^2-1],{z,2}]

erg[z/.raiz]

Manipulate[ Plot[{Cot[z],-Sqrt[(z0/z)^2-1]},{z,0,z0}],{z0,0.9,20}]

Manipulate[ Plot[{Sqrt[-(e+v)/e],-Tan[Sqrt[2 mc2] a/hbarc Sqrt[e+v]]},{e,-v,0},PlotRange->{-1,6}],{{v,v0},0,8000}]

Problema 2.32, gráfico de T

transmissibilidade[e_,a_,v0_,mc2_]:=1/(1+Sin[2a Sqrt[2mc2(e+v0)]/hbarc]^2v0^2/(4e(e+v0)))

Usando os mesmos valores do poço finito para um nucleon

a

v0

mc2

Plot[transmissibilidade[e,a,v0,mc2],{e,0,10v0},PlotRange->{0.9,1}]

Plot[transmissibilidade[e,a,v0,mc2],{e,0,2v0},PlotRange->{0,1}]