ψ0[x_]=(((((m ω)/(h π))))^(1/4))Exp[-((m ω)/(2 h))(x^2)]

ψ1[x_]=FullSimplify[(Sqrt[((m ω)/(2 h))])x ψ0[x]-(Sqrt[(h/(2 m ω))])D[ψ0[x],x]]

ψ2[x_]=FullSimplify[(Sqrt[(1/2)])((Sqrt[((m ω)/(2 h ))])x ψ1[x]-(Sqrt[(h/(2 m ω))])D[ψ1[x],x])]

ψ3[x_]=FullSimplify[(Sqrt[(1/3)])((Sqrt[((m ω)/(2 h ))])x ψ2[x]-(Sqrt[(h/(2 m ω))])D[ψ2[x],x])]

ψ4[x_]=FullSimplify[(Sqrt[(1/4)])((Sqrt[((m ω)/(2 h ))])x ψ3[x]-(Sqrt[(h/(2 m ω))])D[ψ3[x],x])]

h=1
m=1
ω=1

ψ0[x]

Plot[{100ψ0[x]ψ0[x],((x^2)/2)},{x,-8,8},PlotRange->All]

Integrate[ψ1[x]ψ2[x],{x,-∞,∞}]

Ψ[n_,x_]=(1/(Sqrt[(2^n)n!(Sqrt[π])]))Exp[-(x^2)/2]HermiteH[n,x]

ψ4[x]
FullSimplify[Ψ[4,x]]