This script is written in order to study the Spherical Bessel function solutions for an infinitely deep 3-dimensional square well.

Find the 1st zeros of the spherical Bessel function for ℓ = 0 and ℓ = 1.

k0=FindRoot[SphericalBesselJ[0,x],{x,2.5}];
k1=FindRoot[SphericalBesselJ[1,x],{x,4}];
k0=k0[[1,2]]
k1=k1[[1,2]]

Plot the ℓ = 0 and ℓ = 1 spherical Bessel functions such that they share a common intercept, a=1.

Plot[{SphericalBesselJ[0,k0 r],SphericalBesselJ[1,k1 r]},{r,0,2}]

The solutions to the radial equation are:       (Subscript[u, 0])(r) = (Subscript[A, 0]) r (Subscript[𝒿, 0])((Subscript[k, 0]) r)     and      (Subscript[u, 1])(r) = (Subscript[A, 1]) r (Subscript[𝒿, 1])((Subscript[k, 1]) r)       

Find the normalization constants:   (Subscript[A, 0]) and  (Subscript[A, 1]).

n0=Solve[(∫_0^1)(norm0^2) ((SphericalBesselJ[0,k0 r])^2) (r^2)\[DifferentialD]r==1,norm0];
n1=Solve[(∫_0^1)(norm1^2) ((SphericalBesselJ[1,k1 r])^2) (r^2)\[DifferentialD]r==1,norm1];
A0=n0[[2,1,2]]
A1=n1[[2,1,2]]

Plot the probability per unit radius    ((|(Subscript[u, 0])((Subscript[k, 0]) r)|)^2)      and       ((|(Subscript[u, 1])((Subscript[k, 1]) r)|)^2)

Plot[(A0^2) ((SphericalBesselJ[0,k0 r])^2) (r^2),{r,0,1}]

Plot[(A1^2) ((SphericalBesselJ[1,k1 r])^2) (r^2),{r,0,1}]

Plot[{(A0^2) SphericalBesselJ[0,k0 r]^2 r^2,(A1^2) SphericalBesselJ[1,k1 r]^2 r^2},{r,0,1},PlotRange->{0.,2.5}]

Calculate the mean radius of the particle in the infinitely deep 3-dimensional square well for ℓ = 0 and ℓ = 1

mean0=(∫_0^1)(A0^2) ((SphericalBesselJ[0,k0 r])^2) (r^2) r\[DifferentialD]r

mean1=NIntegrate[(A1^2) ((SphericalBesselJ[1,k1 r])^2) (r^2) r,{r,0,1}]