As in version 10.2, there is the NDEigensystem can be used to calculate the eigenstates and eigenvalues of a differential operator.

For example in the harmonic potential case, the even and odd eigen functions can be calculated seperately:

{egnVal1, egnVec1} = 
  NDEigensystem[{-1/2 Laplacian[u[x], {x}] + 1/2 x^2 u[x]}, 
   u[x], {x, 0, 10}, 2];

{egnVal2, egnVec2} = 
  NDEigensystem[{-1/2 Laplacian[u[x], {x}] + 1/2 x^2 u[x], 
    DirichletCondition[u[x] == 0, True]}, u[x], {x, 0, 10}, 2];

And the eigenenergies are:

egnVal = Riffle[egnVal1, egnVal2]
(* {0.500079, 1.50054, 2.5019, 3.5047} *)

And the eigenstates are:

egnVec = Riffle[egnVec1, egnVec2];  
Plot[Evaluate[egnVec], {x, 0, 6}, PlotRange -> All]

which gives the correct wave functions apart from the arbitrary phases.

As in version 10.2, there is the NDEigensystem can be used to calculate the eigenstates and eigenvalues of a differential operator.

For example in the harmonic potential case, the even and odd eigen functions can be calculated using Neumann and Dirichlet boundary condition respectively:

{egnVal1, egnVec1} = 
  NDEigensystem[{-1/2 Laplacian[u[x], {x}] + 1/2 x^2 u[x]}, 
   u[x], {x, 0, 10}, 2];

{egnVal2, egnVec2} = 
  NDEigensystem[{-1/2 Laplacian[u[x], {x}] + 1/2 x^2 u[x], 
    DirichletCondition[u[x] == 0, True]}, u[x], {x, 0, 10}, 2];

And the eigenenergies are:

egnVal = Riffle[egnVal1, egnVal2]
(* {0.500079, 1.50054, 2.5019, 3.5047} *)

And the eigenstates are:

egnVec = Riffle[egnVec1, egnVec2];  
Plot[Evaluate[egnVec], {x, 0, 6}, PlotRange -> All]

which gives the correct wave functions apart from the arbitrary phases.