Harmonic oscillator

$TextStyle = {FontFamily -> Times, FontSize -> 14}

Clear[psi, n, x]
psi[n_, x_] := 
      HermiteH[n, x] Exp[-0.5 x^2]
Energy[n_] := n + 0.5

Clear[rawint]
Do[
rawint[n] = 
        Integrate[
          psi[n, x]^2, {x, -Infinity, Infinity}]; 
Cfac[
          n] = 1/Sqrt[rawint[n]], 
{n, 0, 
        5}]
{Cfac[0], Cfac[5]}

Do[
pl[n] = 
        Plot[Energy[n] + 0.5  Cfac[n] psi[n, x], {x, -5, 
            5}, 
PlotRange -> {All, 
              All}], 
{n, 0, 5}]

Pot = 
      Plot[0.5  x^2, {x, -5, 
          5}, 
PlotRange -> {All, {0, 
              6}}, 
PlotStyle -> 
          RGBColor[1, 0, 0]]
fig = Show[Pot, Table[pl[n], {n, 0, 5}]]

Export["HO.eps", fig]

Time dependent solutions

{n1, n2} = {0, 1}
Psi[x_, t_] := 
      Sqrt[1/2]*psi[n1, x] Exp[-I*Energy[n1] t]
 + 
        Sqrt[1/2]*psi[n2, x] 
          Exp[-I*Energy[n2] t]
Do[Plot[
        Re[Psi[x, t]], {x, -3, 
          3}, 
PlotRange -> {{-3, 3}, {-1.5, 
              1.5}}, 
PlotPoints -> 
          5, 
Epilog -> 
          Text[ToString[t], {2.5, 1.4}]], 
{t, 0, 4  Pi, 
        0.1}]