author: Jorge C. Romão
date : 20/11/2013
email : jorge.romao@tecnico.ulisboa.pt

Problema 8.7

Definition of the Radial functions

R=Function[{n,l,Z,r},(2 Z/(n a0))^(3/2)*Sqrt[(n-l-1)!/(2 n)/((n+l)!)] (2 Z r/(n a0))^l*LaguerreL[n-l-1,2 l+1,2 Z r/(n a0)] E^(-Z r/(n a0))];

Definition of the function given in the problem

psi=Function[{r,alpha},(alpha/Sqrt[Pi])^(3/2) E^(-alpha^2 r^2/2)];

Check Normalization

Print["Norm = ",Integrate[psi[r,alpha]^2 r^2 4Pi,{r,0,Infinity},Assumptions ->alpha>0]]

Calculate Overlap

C100=Integrate[R[1,0,1,r]/Sqrt[4Pi] psi[r,alpha] r^2 4Pi,{r,0,Infinity}, Assumptions->alpha>0]

Print["|C100|^2=",N[C100/. alpha->1/a0]^2]

Plot for alpha=1/a0

Plot[{psi[r,1],R[1,0,1,r]/Sqrt[4Pi]/. a0->1},{r,0,4},AxesLabel->{"r/a0","ψ(r)"},LabelStyle->Directive[14,Bold],PlotStyle->{Thickness[0.0075]},PlotRange->{{0,4},{0,0.6}}]

Plot for alpha=3/a0

Plot[{psi[r,3],R[1,0,1,r]/Sqrt[4Pi]/. a0->1},{r,0,4},AxesLabel->{"r/a0","ψ(r)"},LabelStyle->Directive[14,Bold],PlotStyle->{Thickness[0.0075]},PlotRange->{{0,4},{0,2.5}}]

Plot for alpha=(1/3)/a0

Plot[{psi[r,1/3],R[1,0,1,r]/Sqrt[4Pi]/. a0->1},{r,0,4},AxesLabel->{"r/a0","ψ(r)"},LabelStyle->Directive[14,Bold],PlotStyle->{Thickness[0.0075]},PlotRange->{{0,4},{0,0.6}}]

Define alpha = y/a0 and plot probablity as a function of y*)

C100Fun=Function[y,C100 /. alpha->y/a0];

Plot[C100Fun[y]^2,{y,0.1,5},PlotRange->{0,1},
     AxesLabel->{"y=α a0","|C100|^2"},LabelStyle->Directive[14,Bold],PlotStyle->{Thickness[0.0075]}]