Moreover, we already know that the spherical harmonics are orthonormal [see Equation ([spho])]: that is, \[\oint Y_{l',m'}^{\,\ast}\,Y_{l,m}\,d{\mit\Omega} = \delta_{ll'}\,\delta_{mm'}.\] It, thus, fo...Moreover, we already know that the spherical harmonics are orthonormal [see Equation ([spho])]: that is, \[\oint Y_{l',m'}^{\,\ast}\,Y_{l,m}\,d{\mit\Omega} = \delta_{ll'}\,\delta_{mm'}.\] It, thus, follows that the radial wavefunction satisfies the orthonormality constraint \[\int_0^{\infty} R_{n',l}^\ast\,R_{n,l}\,r^{\,2}\,dr = \delta_{nn'}.\] 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…
