Clear[ψ0,x,m,w,hbar]

(* x=ξ(m w/hbar)^(-1/2); *)
ψ0[x_] = (m w/(Pi hbar))^(1/4)Exp[-m w x^2/(2 hbar)];

wp = 2w;
ψp0[x_] = (m wp/(Pi hbar))^(1/4)Exp[-m wp x^2/(2 hbar)]

ψp1[x_] = Simplify[(1/(2 hbar m wp))^(1/2)(-I hbar/I D[ψp0[x],{x,1}]+m w x ψp0[x]),{hbar>0,m>0,w>0}]

ψp2[x_] = Simplify[(1/(2 hbar m wp))^(1/2)(-I hbar/I D[ψp1[x],{x,1}]+m w x ψp1[x]),{hbar>0,m>0,w>0}]

ψp3[x_] = Simplify[(1/(2 hbar m wp))^(1/2)(-I hbar/I D[ψp2[x],{x,1}]+m w x ψp2[x]),{hbar>0,m>0,w>0}]

Clear[x,m,w,hbar]
m=1;
w=1;
hbar=1;


{
Show[
Plot[m w^2x^2/2-1,{x,-5,5},PlotRange->{-1,1},PlotStyle->Black],
Plot[ψp0[x]^2,{x,-5,5},PlotRange->{-1,1},PlotStyle->Red]
],
Show[
Plot[m w^2x^2/2-1,{x,-5,5},PlotRange->{-1,1},PlotStyle->Black],
Plot[ψp1[x]^2,{x,-5,5},PlotRange->{-1,1},PlotStyle->Green]
],
Show[
Plot[m w^2x^2/2-1,{x,-5,5},PlotRange->{-1,1},PlotStyle->Black],
Plot[ψp2[x]^2,{x,-5,5},PlotRange->{-1,1},PlotStyle->Orange]
],
Show[
Plot[m w^2x^2/2-1,{x,-5,5},PlotRange->{-1,1},PlotStyle->Black],
Plot[ψp3[x]^2,{x,-5,5},PlotRange->{-1,1},PlotStyle->Purple]
]
}

Clear[x,t,m,w,hbar]

ϕ[n_,t_] = Exp[-I (n+1/2)hbar w t/hbar];
Ψ[x_,t_] = ψp0[x]ϕ[0,t] + ψp1[x]ϕ[1,t] + ψp2[x]ϕ[2,t] + ψp3[x]ϕ[3,t];

m=1;
w=1;
hbar=1;

Frames=100;

ExpectX=Quiet[
Table[
Re[NIntegrate[Conjugate[Ψ[x,t]]x Ψ[x,t],{x,-Infinity,Infinity}]],
{t,0,(Frames-1)/Frames 2Pi,2Pi/Frames}
]
];

ListAnimate[
Table[
Plot[Ψ[x,t]Conjugate[Ψ[x,t]],{x,-3,3},PlotRange->{-.5,5},Axes->False,Filling->Axis,Epilog->Line[{{ExpectX[[t Frames/(2Pi)+1]],-.5},{ExpectX[[t Frames/(2Pi)+1]],5}}]],
{t,0,(Frames-1)/Frames 2Pi,2Pi/Frames}
]
]