Clear[r, e, psi, norm, hpsi, ve1, n1, varEnergy,  energy, integrand, gradPsi]
psi[r_] := N E^(-α r)
n1 = 4 Pi Integrate[ r^2  h[r] h[r] , {r, 0, Infinity}]
norm = FullSimplify[ n1,  Re[α]>0]
gradPsi := psi''[r] + 2 psi'[r]/r
hpsi  := FullSimplify[(-ℏ^2/(2m)) gradPsi - e^2 psi[r]/r]
integrand = r^2 hpsi psi[r]
ve1 = 4 Pi Integrate[ integrand, {r, 0, Infinity}]
varEnergy = FullSimplify[ ve1,  Re[α]>0]
energy = varEnergy/norm
de = FullSimplify[D[energy, α]]
Solve[ de== 0, α]

ψ[r_] = FullSimplify[psi[r]/Sqrt[norm], N > 0] /. α->(((e^2) m)/(ℏ^2))
energy /. α->(((e^2) m)/(ℏ^2))