Hydrogen Eigenstates

ψ(R,θ,ϕ) = R(r) (Subscript[Y, LM])(θϕ)

R(r)= F(ρ) Exp[-ρ/2];    ρ = (Sqrt[2|E|]) r   (in atomic units)

(((d^2) F(ρ))/((dρ)^2))+((2/ρ)-1)((d F(ρ))/(dρ))+(((n-1)/ρ)-((L(L+1))/(ρ^2)))F(ρ)=0

n=1,2,....

L=0,1,2,...=n-1,n-2...0

M=L,L-1,...-L

Solutions of F(ρ) giving by generalized Laguerre Polynomials
((Subscript[L, n-L-1])^((2L+1)))(ρ)

Generalized Laguerre polynomials LaguerreL[n, a, x]are related to hydrogen atom wave functions in quantum mechanics.They satisfy the differential equationxy''+(a+1-x)y'+ny=0,and the orthogonality relation(∫_0^∞)(Subscript[L, m]^a)(x)(Subscript[L, n]^a)(x)(x^a)(e^(-x))dx = 0form≠n.TheLaguerre polynomials LaguerreL[n, x]correspond to the special casea=0.

Hpoly[n_,L_,x_]:=LaguerreL[n-L-1,2 L+1,x]

norm[n_,L_]:=Sqrt[ (n-L-1)!/(2n(n+L)!)](2/n)^(3/2)

P[n_,L_,r_]:=norm[n,L]Exp[-r/n] r(2 r/n)^L   Hpoly[n,L, 2 r/n]

Simplify[P[4,3,r]]

Integrate[ P[3,1,r]^2,{r,0,Infinity}]

nq=10;
Plot[P[nq,1,r],{r,0,3nq^2},PlotRange->All]

xyradialdensity[n_,L_,x_,y_]:=(P[n,L,Sqrt[x^2+y^2]])^2
xzradialdensity[n_,L_,x_,z_]:=(P[n,L,Sqrt[x^2+z^2]])^2

nq=1;
plotrange=2nq^2;
DensityPlot[xyradialdensity[nq,0,x,y],{x,-plotrange,plotrange},{y,-plotrange,plotrange},PlotPoints->50,Compiled->True]

(SphericalHarmonicY[1,1,theta,phi] /. {Cos[theta]->x/Sqrt[x^2+y^2],Sin[theta]->y/Sqrt[x^2+y^2],phi->0})^2

" Here we define radial densities projected both on the x-y and x z plane ";

xyangledensity[L_,m_,x_,y_]:=(SphericalHarmonicY[L,m,theta,phi] /. {Cos[theta]->0,Sin[theta]->1,phi->0})^2
xzangledensity[L_,m_,x_,z_]:=(SphericalHarmonicY[L,m,theta,phi] /. {Cos[theta]->z/Sqrt[x^2+z^2],Sin[theta]->x/Sqrt[x^2+z^2],phi->0})^2

" Note; from class and text notes; verify that the charge density is independent
on azimuthal angle phi; however charge density does depend on m since the
the associated Legendre polynomials P[L,m] have m-dependence ";

" Total charge density on two planes";

xyTwoDensity[n_,L_,m_,x_,y_]:=xyradialdensity[n,L,x,y] xyangledensity[L,m,x,y]
xzTwoDensity[n_,L_,m_,x_,z_]:=xzradialdensity[n,L,x,z] xzangledensity[L,m,x,z]

" Some plots of the n=2, l=1, m=-1,0,1 states projected on the
   xy plane "; 

nq=2;
plotrange=2nq^2;
"for m=-1";
DensityPlot[xyTwoDensity[nq,1,-1,x,y],{x,-plotrange,plotrange},{y,-plotrange,plotrange},PlotPoints->40,Compiled->True]

"for m=0";
DensityPlot[xyTwoDensity[nq,1,0,x,y],{x,-plotrange,plotrange},{y,-plotrange,plotrange},PlotPoints->40,Compiled->True]

"for m=1";
DensityPlot[xyTwoDensity[nq,1,1,x,y],{x,-plotrange,plotrange},{y,-plotrange,plotrange},PlotPoints->40,Compiled->True]

" The same plots of the n=2, l=1, m=-1,0,1 states projected on the
   xz plane "; 

nq=2;
plotrange=2nq^2;
"for m=-1";
DensityPlot[xzTwoDensity[nq,1,-1,x,y],{x,-plotrange,plotrange},{y,-plotrange,plotrange},PlotPoints->40,Compiled->True]

"for m=0";
DensityPlot[xzTwoDensity[nq,1,0,x,y],{x,-plotrange,plotrange},{y,-plotrange,plotrange},PlotPoints->40,Compiled->True]

"for m=1";
DensityPlot[xzTwoDensity[nq,1,1,x,y],{x,-plotrange,plotrange},{y,-plotrange,plotrange},PlotPoints->40,Compiled->True]