ContourPlot[Re[SphericalHarmonicY[5, 2, t,p]],{p,-Pi,Pi},{t,0,Pi},AspectRatio->Automatic]
4 azi nodes; 3 zonal nodes

ContourPlot[Re[SphericalHarmonicY[6, 2, t,p]],{p,-Pi,Pi},{t,0,Pi},AspectRatio->Automatic]
4 azi nodes; 4 zonal nodes

ContourPlot[Re[SphericalHarmonicY[6, 3, t,p]],{p,-Pi,Pi},{t,0,Pi},AspectRatio->Automatic]
6 azi nodes; 3 zonal nodes

# azi = 2*m; #zonal=(l-m)

ContourPlot[Abs[SphericalHarmonicY[5, 2, theta,phi]],{phi,-Pi,Pi},{theta,0,Pi},AspectRatio->Automatic]
Abs[Exp[I m phi]]=1

R[n_,l_,r_]=Sqrt[2] SphericalBesselJ[l,BesselJZero[l+1/2, n] r]/Abs[SphericalBesselJ[l+1,BesselJZero[l+1/2, n]]]

Plot[R[3,2,r],{r,0,1}]
2 nodes


Plot[R[4,2,r],{r,0,1}]
3 nodes

Plot[R[2,2,r],{r,0,1}]
1 node

n-1=#nodes

Plot[R[2,1,r],{r,0,1}]
starts linear

Plot[R[2,0,r],{r,0,1}]
starts constant

Plot[R[2,2,r],{r,0,1}]   
starts quadratic

starts r^L
L=0 for R(0) not zero

the volume element is r^2 dr dOmega; dOmega goes with SphericalHarmonicY; r^2 sticks with dr

Table[Integrate[R[3,2,r] R[n,2,r] r^2,{r,0,1}],{n,1,4}]
Out[12]= {0, 0, X, 0}

Table[Integrate[R[3,0,r] R[n,0,r] r^2,{r,0,1}],{n,1,4}]
Out[13]= {0, 0, 1, 0}

Table[N[Integrate[R[n,2,r] R[n,2,r] r^2,{r,0,1}]],{n,1,4}]
Out[14]= {1., 1., 1., 1.}

Table[Integrate[R[n,0,r] R[n,0,r] r^2,{r,0,1}],{n,1,4}]
Out[15]= {1, 1, 1, 1}

1000/9 Integrate[ r^2,{r,0,.3}]
Out[16]= 1.

Table[Sqrt[1000/9] Integrate[R[n,0,r] r^2,{r,0,.3}],{n,1,100}] 
Out[50]= {0.385218, 0.579067, 0.503144, 0.232427, -0.0604163, -0.216604, 
max is at n=2

res=%^2
Out[51]= {0.148393, 0.335319, 0.253154, 0.0540222, 0.00365013, 0.0469171, 
13th 0.0114285 is last above 1%

Sum[res[[i]],{i,1,100}]                                                

Out[52]= 0.989969

Sum[res[[i]],{i,1,50}]                                                 

Out[53]= 0.980137


for SHO splittings are same (energies evenly spaced); not so for infinte spherical square well

e.g., (3s-2s)/(2s-1s)= (BesselJZero[1/2, 3]^2-BesselJZero[1/2, 2]^2)/(BesselJZero[1/2, 2]^2-BesselJZero[1/2, 1]^2)
Out[18]= 1.66667

for SHO 1p at midpoint of 2s&1s; not so for infinte spherical square well
