Spherical Bessel Functions

Plot[{SphericalBesselJ[0,x],SphericalBesselJ[1,x],SphericalBesselJ[2,x]},{x,0,20},PlotRange-> {All,All},PlotStyle->{{Thickness[0.005],Hue[0.1]},{Thickness[0.005],Hue[0.3]},{Thickness[0.005],Hue[0.6]}}]

Plot[{SphericalBesselY[0,x],SphericalBesselY[1,x],SphericalBesselY[2,x]},{x,0,20},PlotRange-> {All,{-1,0.5}},PlotStyle->{{Thickness[0.005],Hue[0.1]},{Thickness[0.005],Hue[0.3]},{Thickness[0.005],Hue[0.6]}}]

Free Particle Wave Functions

freeparticlewf[k_,l_,m_,r_,θ_]=SphericalBesselJ[l,k r]  SphericalHarmonicY[l,m,θ,0]

DensityPlot[freeparticlewf[3,0,0,Sqrt[x^2+y^2],ArcTan[x/y]]^2,{x,-10,10},{y,-10,10},PlotPoints->100,Mesh->False,PlotRange->{0,0.005}]

DensityPlot[freeparticlewf[3.5,1,0,Sqrt[x^2+y^2],ArcTan[x/y]]^2,{x,-10,10},{y,-10,10},PlotPoints->200,Mesh->False, PlotRange->{-0,0.005}]

DensityPlot[freeparticlewf[3,2,0,Sqrt[x^2+y^2],ArcTan[x/y]]^2,{x,-10,10},{y,-10,10},PlotPoints->200,Mesh->False, PlotRange->{-0,0.005}]

(* Manipulate[
DensityPlot[freeparticlewf[k,0,0,Sqrt[x^2+y^2],ArcTan[x/y]]^2,{x,-10,10},{y,-10,10},Mesh->False,PlotPoints->80,PlotRange->{0,0.005}],
{k,1,5}] *)

Particle in a Spherical Cavity

wf1[n_,l_]:=Module[{t,g},
t=N[BesselJZero[l+(1/2),n]];
g=Which[x^2+y^2<1,SphericalBesselJ[l,t Sqrt[x^2+y^2]]^2  SphericalHarmonicY[l,0,ArcTan[x/y],0]^2,True,0];
DensityPlot[g,{x,-1.2,1.2},{y,-1.2,1.2},PlotPoints->100,Mesh->False,PlotRange->Automatic,Epilog->{Circle[{0,0},1]}]
]

wf1[1,0]

wf1[2,0]

wf1[3,0]

wf1[1,1]

wf1[1,2]

Manipulate[Plot[SphericalBesselJ[0,k x],{x,0,4},PlotRange->{-0.5,1}],{k,1,10}]