ClearAll;
n=3;
HermiteH[n,x]
Plot[HermiteH[n,x],{x,-10,10}]

a=1;  (*** a= (Sqrt[((m ω)/ℏ)])****)
n=10;
x0=N[(Sqrt[2 (((n+1/2))/(a^2))])]
p1=Plot[(((((a^2)/(π (2^(2n)) ((Factorial[n])^2)))))^(1/2)) Exp[- (a^2) (x^2)] ((HermiteH[n, a x])^2) ,{x,-1.3 x0,1.3 x0}, PlotRange->All, PlotStyle->Blue];
p2=Plot[(0.3/(Sqrt[(x0^2)-(x^2)])),{x,-x0, x0}, PlotStyle->Red]

Show[p1, p2, GridLines->{{{-x0, Black},{x0, Black}},{0}}]