Moreover, we already know that the spherical harmonics are orthonormal [see Equation ([spho])]: that is, ∮Y∗l′,m′Yl,mdΩ=δll′δmm′. It, thus, fo...Moreover, we already know that the spherical harmonics are orthonormal [see Equation ([spho])]: that is, ∮Y∗l′,m′Yl,mdΩ=δll′δmm′. It, thus, follows that the radial wavefunction satisfies the orthonormality constraint ∫∞0R∗n′,lRn,lr2dr=δnn′. The first few radial wavefunctions for the hydrogen atom are listed below: \[\begin{aligned} R_{1,0}(r)&= \frac{2}{a_0^{\,3/2}}\,\exp\left(-\frac{r}{a_0…
