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I'm trying to solve the Schrodinger equation and having difficulties to define a limitless region because the problem has the Dirichlet conditions at infinity. Maybe I don't need such a region and should merely to define that condition correctly but I don't know how to implement it

h = 1; m = 1; V[r_] := -1/Sqrt[ r.r];
\[ScriptCapitalL] = -(h^2/(2 m)) \!\(
\*SubsuperscriptBox[\(\[Del]\), \({x, y, z}\), \(2\)]\(f[x, y, 
      z]\)\) + V[{x, y, z}] f[x, y, z];
n = 3;


 A = ImplicitRegion[
     -\[Infinity] < x  < \[Infinity] && -\[Infinity] < 
     y < \[Infinity]  &&  -\[Infinity] < z < \[Infinity] , {x, y, 
        z}];
    {vals, funs} = 
      NDEigensystem[{\[ScriptCapitalL], 
        DirichletCondition[f[x, y, z] == 0, True]}, 
       f, {x, y, z} \[Element] A, n] ;
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  • $\begingroup$ This code has this message NDEigensystem::femtemnbb: The bounds for ImplicitRegion[-\[Infinity]<x<\[Infinity]&&-\[Infinity]<y<\[Infinity]&&-\[Infinity]<z<\[Infinity],{x,y,z}] are {{-\[Infinity],\[Infinity]},{-\[Infinity],\[Infinity]},{-\[Infinity],\[Infinity]}}. Unless finite numeric bounds are specified, the mesh generation will constrain the region to have finite bounds. $\endgroup$ – Alex Trounev Oct 10 '18 at 16:21
  • $\begingroup$ @AlexTrounev, thank you again for an answer. But I think I need to define boundary conditions to get the valid solution, because I'd like to have the solution for an atom in infinite region where the ground state energy is -1/2. And mesh for such a region will have a lot of memory consumption I suppose $\endgroup$ – James Flash Oct 10 '18 at 16:31
  • $\begingroup$ You can’t get the right solution for a hydrogen atom just by putting an electron in a box and setting a potential. It is necessary to limit the binding energy so that the electron is not free in the box. $\endgroup$ – Alex Trounev Oct 10 '18 at 16:39
  • $\begingroup$ @AlexTrounev, do you mean to use assumption that E < 0? Ok, I'll try it $\endgroup$ – James Flash Oct 10 '18 at 16:44
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To limit the binding energy, proceed as follows:

h = 1; m = 1; V[r_] := 1 - 1/Sqrt[ r.r]

\[ScriptCapitalL] = -(h^2/(2 m)) \!\(
\*SubsuperscriptBox[\(\[Del]\), \({x, y, z}\), \(2\)]\(f[x, y, 
      z]\)\) + V[{x, y, z}] f[x, y, z];
 d = 10; n = 3;
A = ImplicitRegion[-d <= x <= d && -d <= y <= d && -d <= z <= d, {x, 
    y, z}];


 {vals, funs} =  
  NDEigensystem[{\[ScriptCapitalL], 
    DirichletCondition[f[x, y, z] == 0, True]}, 
   f, {x, y, z} \[Element] A, n] ;

Here we get three eigenvalues, the first of which is close to -1/2, and the other two to -1/8:

 vals - 1

Out[]= {-0.479657, -0.121793, -0.121756}

The inaccuracy of the definition of eigenvalues is due to the influence of the box boundaries. The distribution of the square of the amplitude of the wave function approximately corresponds to n = 1 and 2. In the second case, the angular momentum l > 0.

Table[ContourPlot[
  Evaluate[funs[[i]][x, y, 0]], {x, -d, d}, {y, -d, d}, 
  PlotRange -> All, PlotLabel -> vals[[i]] - 1, Contours -> 20, 
  PlotLegends -> Automatic], {i, Length[vals]}]

fig1

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  • $\begingroup$ Thank you very much. I appreciate your help $\endgroup$ – James Flash Oct 10 '18 at 17:54
  • 1
    $\begingroup$ You're welcome! I think that in this model you can get a very interesting result concerning the splitting of energy levels for a given perturbation at the boundary. This is evident in the level of E = -1 / 8. $\endgroup$ – Alex Trounev Oct 10 '18 at 18:04

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