Plot[HermiteH[1,x],{x,-2,2}]

Plot[HermiteH[4,x],{x,-2,2}]

(* Así la función no diverge si n es un entero*)
Plot[Exp[-(x^2)/2]HermiteH[50,x],{x,-20,20}]

(* Pero cuando n no es un entero la función diverge *)
Plot[Exp[-(x^2)/2]HermiteH[(Sqrt[2]),x],{x,-20,20}]

ϕ[n_,x_]:=(Sqrt[(1/(2n!))])(((((m ω)/(π ℏ))))^(1/4))Exp[-((m ω (x^2))/(2 ℏ))]HermiteH[n,(Sqrt[m ω /ℏ])x];
funs=Table[3 n+ϕ[n,x]/.{m->1,ℏ->1,ω->1},{n,1,4}]
V=1/2 m (ω^2)(x^2)/.{m->1,ℏ->1,ω->1}

Plot[{funs ,V},{x,-10,10}]

funs=Table[3 n+ϕ[n,x]/.{m->1,ℏ->1,ω->1},{n,5,10}]
Plot[{funs,V},{x,-10,10}]