How to make a 3d plot out of a 2d figure?

Question

Hello guys I am trying to make a 3d plot out of a 2d figure, but things are going sideways. My commands are

$Assumptions = Element[A | \[Epsilon] | m | Subscript[\[CapitalOmega],0] | \[Mu] | \ 
[Nu], Reals] && A > 0 && \[Epsilon] > 0 &&   0 < \[Nu] < \[Pi] && m > 0 &&  Subscript[\ 
[CapitalOmega], 0] > 0 && \[Mu] > 0;

(Variables used)

Np = 10^5;
\[Epsilon] =  1.0 10^(-1);
m = 6.49*10^(-26);
Subscript[a, s] = 9*10^(-6);
A = 5*10^(-6);
\[HBar] = 1.054571817*10^(-34) ;
Subscript[\[CapitalOmega], 0] =  2 \[Pi] 200;
Subscript[g, 2 D] =   Sqrt[((m Subscript[\[CapitalOmega], 0])/(2 \[Pi] \[HBar]))]*(4 \ 
[Pi] \[HBar]^(2) Subscript[a, s])/  m
\[Mu] = 9.0 10^(-11);

(Defining my function-Angular part)

Subscript[\[Xi], tf] =  Sqrt[(\[Mu] - (\[HBar] \[Epsilon])/(2 m) 
(Cos[\Nu]])^2)/Subscript[g, 2 D]]

(Ploting the angular part)

Plot[{Abs[Subscript[\[Xi], tf]^{2}]}, {\[Nu], 0, \[Pi]}, 
AxesStyle -> Arrowheads[{0.0, 0.03}], 
LabelStyle -> Directive[Black, 28, Bold], 
AxesLabel -> {Style[\[Nu], 28, Black], 
Style[Abs[(Subscript[\[Xi], TF])^2], 28, Black]}, 
PlotLegends -> Style[Abs[Subscript[\[Xi], TF]^2], 28, Black], 
PlotRange -> Automatic, PlotStyle -> {{Blue, Thick}}, 
ImageSize -> Large, 
Ticks -> {{0, \[Pi]/4, \[Pi]/2, 3*\[Pi]/4, \[Pi]}, Automatic}]

(Now the full funciton with radial part)

\[Psi][r_, \[Nu]_] = 
Abs[(((m Subscript[\[CapitalOmega], 0])/(\[Pi] \[HBar]))^(1/
     4) Exp[-((m Subscript[\[CapitalOmega], 0] )/(
     2 \[HBar]))*(r - A)^2] Subscript[\[Xi], tf])^2];

escalaTF = A + A/3;

(Density plot of full function)

DensityPlot[\[Psi][Sqrt[x^2 + (1 + \[Epsilon]) z^2], 
ArcCos[(z *Sqrt[1 + \[Epsilon]])/
Sqrt[x^2 + (1 + \[Epsilon]) z^2]]], {x, -escalaTF, 
escalaTF}, {z, -escalaTF, escalaTF}, Exclusions -> None, 
PlotPoints -> 100, ColorFunction -> "BlueGreenYellow", 
LabelStyle -> {23}, PlotLegends -> Automatic] 

(UP UNTIL THIS POINT IM GETTING WHAT I NEED, NOW THE PROBLEM BEGINS)

(I want the full 3d vizualization of my sphere, so I try the commands)

\[Psi][x_, z_] = \[Psi][Sqrt[x^2 + (1 + \[Epsilon]) z^2], 
ArcCos[(z *Sqrt[1 + \[Epsilon]])/Sqrt[x^2 + (1 + \[Epsilon]) z^2]]]\[Psi][x_, z_] = \ 
[Psi][Sqrt[x^2 + (1 + \[Epsilon]) z^2], 
ArcCos[(z *Sqrt[1 + \[Epsilon]])/Sqrt[x^2 + (1 + \[Epsilon]) z^2]]]

Raio = A^(2);

a[x_, z_] = 
Piecewise[{{\[Psi][x, Sqrt[(Raio - x^2)/(1 + \[Epsilon])]], 
x^2 + (1 + \[Epsilon]) z^2 < Raio}, {0, 
x^2 + (1 + \[Epsilon]) z^2 > Raio}}]

DensityPlot[
a[x, z], {x, -escalaTF, escalaTF}, {z, -escalaTF, escalaTF}, 
Exclusions -> None, ColorFunction -> "BlueGreenYellow", 
PlotRange -> Automatic, PlotPoints -> 100, LabelStyle -> {23}, 
PlotLegends -> Automatic] 

So the idea is that in spherical coordinates the nu angle goes from 0 to pi in the z direction, and the phi angle gos from 0 to 2 pi in the x-y plane. I want to plot the full sphere seen from the ourside so i try to rotate the figure of the shell on the phi angle from 0 to 2 pi.

I hope you can note this is not the full sphere seen from the outside. It should have only a small blue circle on the south and north poles, and a yellow horizontal band at the equator and not a blue vertical band. I don't understand what I did wrong. This is not the usual spherical coordinates, since I added a distortion on the z direction to make an ellipsoid if wanted.

I also tried a 3d plot

Subscript[\[Psi], S][
r_, \[Nu]_] = ((
 m Subscript[\[CapitalOmega], 0])/(\[Pi] \[HBar]))^(1/4) Exp[-((
  m Subscript[\[CapitalOmega], 0] )/(
  2 \[HBar]))*(r - A)^2] Subscript[\[Xi], tf];

Subscript[\[Psi], S][x_, y_, z_] = 
Subscript[\[Psi], S][Sqrt[x^2 + y^2 + (1 + \[Epsilon]) z^2], 
ArcCos[(z *Sqrt[1 + \[Epsilon]])/
Sqrt[x^2 + y^2 + (1 + \[Epsilon]) z^2]]]

Raio = A^(2);

a[x_, y_, z_] = 
Piecewise[{{Subscript[\[Psi], S][x, 
 Sqrt[(Raio - x^2 - y^2)/(1 + \[Epsilon])]], 
x^2 + y^2 + (1 + \[Epsilon]) z^2 < Raio}, {0, 
x^2 + y^2 + (1 + \[Epsilon]) z^2 > Raio}}]

DensityPlot3D[
a[x, y, z], {x, -escalaTF, escalaTF}, {y, -escalaTF, 
escalaTF}, {z, -escalaTF, escalaTF}, 
ColorFunction -> "BlueGreenYellow", PlotRange -> Automatic, 
PlotPoints -> 50, LabelStyle -> {23}, PlotLegends -> Automatic]

But this has the same problem of the blue band going from north to south as in the other figure.

Answer 1

Is this what you're after?

(* from the question *)
Np = 10^5;
ϵ = 1.0 10^(-1);
m = 6.49*10^(-26);
Subscript[a, s] = 9*10^(-6);
A = 5*10^(-6);
ℏ = 1.054571817*10^(-34);
Subscript[Ω, 0] = 2 π 200;
Subscript[g, 2 D] = 
  Sqrt[((m Subscript[Ω, 
         0])/(2 π ℏ))]*(4 π ℏ^(2) Subscript[a, 
       s])/m;
μ = 9.0 10^(-11);

Subscript[ξ, tf] = 
  Sqrt[(μ - (ℏ ϵ)/(2 m) (Cos[ν])^2)/
    Subscript[g, 2 D]];

ψ[r_, ν_] = 
  Abs[(((m Subscript[Ω, 0])/(π ℏ))^(1/
         4) Exp[-((m Subscript[Ω, 0])/(2 ℏ))*(r - 
           A)^2] Subscript[ξ, tf])^2];

escalaTF = A + A/3;

(* plotting code *)
DensityPlot3D[ψ[Sqrt[x^2 + y^2 + z^2], 
  ArcTan[z, Sqrt[x^2 + y^2]]], {x, -escalaTF, 
  escalaTF}, {y, -escalaTF, escalaTF}, {z, -escalaTF, escalaTF}, 
 ColorFunction -> "BlueGreenYellow", PlotRange -> Automatic, 
 LabelStyle -> {23}, PlotLegends -> Automatic]