Finding the bound states in a finite spherical well.

mc2=938.;     "mass of a nucleon";
ℏc=197.;        "MeV•fm";
v=70;               "depth of the potential well";
a=3.5 ;            "radius of the nucleus"  ;

Focussing on the ℓ = 0 spherical Bessel function, find the minimum potential energy v0 to create the first 3 bound states (i.e., 𝓃 = 1, 2 and 3).

v0=.       "Let the depth of the potential well vary from 0 -> 150 MeV.";
Eb=.
Manipulate[Plot[{Tan[(Sqrt[((2 mc2)/(((ℏc))^2))(v0-Eb) (a^2)])],-(Sqrt[(v0/Eb)-1])},{Eb,0,100},PlotRange->{-20,20}],{v0,1,100}]

Provide the depth of the potential energy well ``v0’’ and the starting value for Eb in order to find the intersection point and the value for Eb.

v0=70.0;
Eb=.
EbStart=22;
Plot[{Tan[(Sqrt[((2 mc2)/(((ℏc))^2))(v0-Eb) (a^2)])],-(Sqrt[(v0/Eb)-1])},{Eb,0,90},PlotRange->{-20,20}]
FindRoot[Tan[(Sqrt[((2 mc2)/(((ℏc))^2))(v0-Eb) (a^2)])]==-(Sqrt[(v0/Eb)-1]),{Eb,EbStart}]
Eb=%[[1,2]];

Find the normalization constant   (Subscript[𝒸, \[ScriptOne]])  and plot the probabilty density |𝓊(r)((|)^2)

v0
Eb
u=.

k=(Sqrt[((2 mc2(v0-Eb))/((ℏc)^2))]);
κ=(Sqrt[((2 mc2 Eb)/((ℏc)^2))]);
const1=Solve[(c1Sq/(k^2))((∫_0^a)((Sin[k r])^2)\[DifferentialD]r + ((Sin[k a])^2) (∫_a^∞)(E^(-2 κ (r-a)))\[DifferentialD]r)==1,c1Sq];
c1=(Sqrt[const1[[1,1,2]]]);
Print[" a = ",a,"      k = ",k,"     κ = ",κ,"     c1 = ",c1]
u[r_]:=({{\[Piecewise], ({{(c1/k) Sin[k r], 0≤r≤a}, {(c1/k) Sin[k a] (E^(-κ (r-a))), a≤r<∞}})}})
u[r]
Plot[((u[r])^2),{r,0,8}]

norm=(∫_0^∞)((u[r])^2)\[DifferentialD]r

(Sqrt[(∫_0^∞)u[r] (r^2) u[r]\[DifferentialD]r-((((∫_0^∞)u[r] r u[r]\[DifferentialD]r))^2)])

(∫_0^∞)u[r] r u[r]\[DifferentialD]r