α = ((k*μ)/ℏ^2)^0.5

NormConst[v_]:= 1/(2^v*Factorial[v])^(1/2)*(α/Pi)^(1/4)

Psi[x_,v_]:=NormConst[v]*HermiteH[v,α^(1/2)*x]*Exp[-α*x^2/2]

Integrate[Psi[x,1] x  Psi[x,1] ,{x,-Infinity, Infinity} , Assumptions-> {k∈ Reals, μ ∈ Reals, ℏ∈ Reals,  k > 0,  μ>0, ℏ>0}]

Integrate[Psi[x,1] x x Psi[x,1] ,{x,-Infinity, Infinity} , Assumptions-> {k∈ Reals, μ ∈ Reals, ℏ∈ Reals,  k > 0,  μ>0, ℏ>0}]

Integrate[Psi[x,1] ( -1 ℏ^2 D[ Psi[x,1],x,x]) ,{x,-Infinity, Infinity} , Assumptions-> {k∈ Reals, μ ∈ Reals, ℏ∈ Reals,  k > 0,  μ>0, ℏ>0}]

Sqrt[%5] Sqrt[%6]