Solve quantum mechanics 1D box using NDEigensystem

Question

L = 2   
V[x_] = 
  Piecewise[{
    {Infinity, x <= -L}, 
    {0, -L < x < L},
    {Infinity, x >= L}
  }];

{vals, funs} = 
  NDEigensystem[
    {-Laplacian[u[x], {x}] + V[x]*u[x],
     DirichletCondition[u[x] == 0, True]},
    u[x], {x, -2, 2}, 10,
    Method -> {"SpatialDiscretization" -> {"FiniteElement", {"MeshOptions" -> {MaxCellMeasure -> 0.01}}}}
  ]

Where have I made a mistake?

Answer 1

You don't need to specify the potential beyond the Dirichlet boundaries:

L = 2;

V[x_] = 0;  (*  will only be used in [-L,+L]  *)

{vals, funs} = 
  NDEigensystem[{-Laplacian[u[x], {x}] + V[x]*u[x], 
    DirichletCondition[u[x] == 0, True]},
    u[x], {x, -L, L}, 10, 
    Method -> {"SpatialDiscretization" -> {"FiniteElement", {"MeshOptions" -> {MaxCellMeasure -> 0.01}}}}]

The eigenvalues are correct numerically:

vals - Table[(n*π/4)^2, {n, 10}]

(*    {4.27502*10^-12, 2.12463*10^-10, 2.37989*10^-9,
       1.33551*10^-8, 5.09357*10^-8, 1.5208*10^-7,
       3.83468*10^-7, 8.54395*10^-7, 1.732*10^-6,
       3.25887*10^-6}                                    *)

Answer 2

Do not use Infinity for a numerical value. Use e.g. 10^6:

L = 2
V[x_] = Piecewise[{{10^6, x <= -L}, {0, -L < x < L}, {\[Infinity], 
     x >= L}}];
{vals, funs} = 
 NDEigensystem[{-Laplacian[u[x], {x}] + V[x]*u[x], 
   DirichletCondition[u[x] == 0, True]}, u[x], {x, -2, 2}, 10, 
  Method -> {"SpatialDiscretization" -> {"FiniteElement", \
{"MeshOptions" -> {MaxCellMeasure -> 0.01}}}}]

The first 6 eigenfunction look like:

Plot[Evaluate@funs[[;; 6]], {x, -2, 2}]