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Question duplicated from here https://mathematica.stackexchange.com/questions/287433/why-if-the-potential-is-different-from-the-coulomb-one-but-has-spherical-symme and here https://physics.stackexchange.com/questions/771468/why-if-the-potential-is-different-from-the-coulomb-one-but-has-spherical-symme

I have found the eigenvalues of the following systems: $H=-\frac{1}{2}\Delta+V_1$ and $H=-\frac{1}{2}\Delta+V_2$, using NDEigensystem by Wolfram Mathematica.

In the first case the potential is Coulomb one ($V_1=-\frac{1}{\sqrt{\rho^2+z^2}}$ (is written in a cylindrical coordinate system)). The second potential has a more complex structure $V_2=-\frac{1}{\sqrt{\rho^2+z^2}}-\frac{1}{2\sqrt{\rho^2+z^2}}e^{\frac{1}{2\sqrt{\rho^2+z^2}}}$.

The codes are written in Wolfram Mathematica. In the both codes I renamed $\rho≡r$

1.

ClearAll["Global`*"]
rmax = 20;
zmax = 20;

V1[r_, z_] := -1/Sqrt[r^2 + z^2]
{valsc, funsc} = 
  NDEigensystem[{(-1/2*
        Laplacian[\[Psi][r, z], {r, \[Theta], z}, "Cylindrical"] + 
       V1[r, z]*\[Psi][r, z]) + \[Psi][r, z]*0.5}, \[Psi][r, z], {r, 0,
     rmax}, {z, -zmax, zmax}, 20, 
   Method -> {"SpatialDiscretization" -> {"FiniteElement", \
{"MeshOptions" -> {"MaxCellMeasure" -> 0.05}}}, 
     "Eigensystem" -> {"Arnoldi", "MaxIterations" -> 10000}}];
Sort[valsc] - 0.5

20 first eigenvalues obtained from the code 1:

{-0.503427, -0.125434, -0.125001, -0.0591382, -0.0575661, \
-0.0567415, -0.0366165, -0.0348045, -0.0331543, -0.0305338, \
-0.0194174, -0.011569, -0.00383252, -0.000633831, 0.0115625, \
0.0130426, 0.0228641, 0.0287679, 0.0380814, 0.0405388}

As follows from the general theory, the levels are degenerate:

-0.503427 — 1s state
-0.125434, -0.125001 — 2s and 2p states
-0.0591382, -0.0575661, -0.0567415 — 3s, 3p and 3d states
...

2.

ClearAll["Global`*"]
rmax = 20;
zmax = 20;

V2[r_, z_] := -1/Sqrt[r^2 + z^2] - 
  1/2/Sqrt[r^2 + z^2]*Exp[-Sqrt[r^2 + z^2]]
{valsc1, funsc1} = 
  NDEigensystem[{(-1/2*
        Laplacian[\[Psi][r, z], {r, \[Theta], z}, "Cylindrical"] + 
       V2[r, z]*\[Psi][r, z]) + \[Psi][r, z]*0.5}, \[Psi][r, z], {r, 
    0, rmax}, {z, -zmax, zmax}, 20, 
   Method -> {"SpatialDiscretization" -> {"FiniteElement", \
{"MeshOptions" -> {"MaxCellMeasure" -> 0.05}}}, 
     "Eigensystem" -> {"Arnoldi", "MaxIterations" -> 10000}}];
Sort[valsc1] - 0.5

20 first eigenvalues obtained from the code 2:

{-0.816105, -0.152359, -0.134395, -0.0649383, -0.0597383, \
-0.0570375, -0.0381872, -0.0375616, -0.0340771, -0.0326892, \
-0.0220107, -0.0128179, -0.00388086, -0.00246523, 0.0106794, \
0.011968, 0.0221767, 0.0277981, 0.0328003, 0.0358925}

As can be seen in this case, the eigenvalues are different — the levels are not degenerate:

-0.816105 — 1s state
-0.152359 — first excited state
-0.134395 — second excited state
...

Question:
Why are the levels not degenerate in the second case? Like the first potential ($V_1$), the second potential ($V_2$) has spherical symmetry, hence the system is spherically symmetric. Shouldn't any spherically symmetric system give levels that are degenerate in the orbital quantum number?

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  • $\begingroup$ I’m voting to close this question because its a cross site duplicate that has an answer on Physics SE $\endgroup$
    – Tyberius
    Jul 10, 2023 at 17:55

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