[UPSC IFoS 2018] Angular momentum, spin etc (Q286, 5+15=20M)

The radial part of Schrödinger wave function for hydrogen atom for spherical symmetric potential V(r) = -\frac{e^2}{4\pi \varepsilon_0 r} is given as : \frac{1}{r^2}\frac{d}{dr}\left(r^2 \frac{dR}{dr}\right) + \frac{2\mu}{\hbar^2}\left[ E - V(r) - \frac{\hbar^2}{2\mu}\frac{l(l+1)}{r^2}\right] R = 0, where \mu = \frac{mM}{m+M} is reduced mass and m and M are mass of electron and proton respectively. (i) Obtain the form of the above equation for ground state electron of hydrogen atom. (ii) Also starting with a trial wave function for the radial equation, R = A e^{-r/a_0}, for the l = 0 state, find the expressions of energy E and orbital radius for the ground state of hydrogen atom.