Simple Harmonic Oscillator - The Answers

Define (Subscript[ψ, n])(x) .

ψ[n_,x_]:=((((1/π))^(1/4))/(Sqrt[(2^n)n!]))(E^(-(x^2)/2)) HermiteH[n,x];

Graph (Subscript[Ψ, n])(x,0).

Manipulate[
Show[
Plot[
ψ[n,x],{x,-10,10},
PlotRange->{{-10,10},{-0.7,0.8}},
Ticks->{  {{10,""}},{ {-0.6,""},{-0.4,""},{-0.2,""},{0.2,""},{0.4,""},{0.6,""},{0.8,""}}},
PlotStyle->{Orange,Thickness[0.005]},
AxesLabel->{x,"ψ(x,0)"},
ImageSize->800,
LabelStyle->({FontSize->16,FontFamily->"Times"})
],
ListLinePlot[
{{-(Sqrt[2n+1]),-0.7},{-(Sqrt[2n+1]),+0.8}},
PlotStyle->{Blue,Thickness[0.001],Dotted}
],
ListLinePlot[
{{(Sqrt[2n+1]),-0.7},{(Sqrt[2n+1]),+0.8}},
PlotStyle->{Blue,Thickness[0.001],Dotted}
]
],
{{n,0,"n"},Range[0,40,1],Setter}
]

Graph ((||Ψ(x)||)^2).

Manipulate[
Show[
Plot[
Conjugate[ψ[n,x]]*ψ[n,x],{x,-10,10},
PlotRange->{{-10,10},{-0.1,0.6}},
Ticks->{  {{10,""}},{ {0.1,""},{0.2,""},{0.3,""},{0.4,""},{0.5,""},{0.6,""}}},
PlotStyle->{Orange,Thickness[0.005]},
AxesLabel->{x,"\[LeftBracketingBar]Ψ(\[RightBracketingBar]^2)"},
ImageSize->800,
LabelStyle->({FontSize->16,FontFamily->"Times"})
],
ListLinePlot[
{{-(Sqrt[2n+1]),0},{-(Sqrt[2n+1]),+0.6}},
PlotStyle->{Blue,Thickness[0.001],Dotted}
],
ListLinePlot[
{{(Sqrt[2n+1]),0},{(Sqrt[2n+1]),+0.6}},
PlotStyle->{Blue,Thickness[0.001],Dotted}
]
],
{{n,0,"n"},Range[0,40,1],Setter}
]