Question

Obtain the most probable value and expectation value of r for the ground state of a
hydrogen atom.

Answer 1

The wave function of an electron in the ground state of hydrogen atom

$$\psi=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}$$

The probability density

$$\rho=4\pi r^2|\psi|^2=\frac{4r^2}{a_0^3}e^{-2r/a_0}$$

$$\frac{d\rho}{dr}=\frac{8r}{a_0^3}\left[1-\frac{r}{a_0}\right]e^{-2r/a_0}=0$$

So, the most probable value value of r 

$$r_{\rm{mp}}=a_0$$

The expectation value of r

$$<{\hat r}>=\int_0^{\infty} r \rho dr=\int_0^{\infty}\frac{4r^3}{a_0^3}e^{-2r/a_0}=\frac{3}{2}a_0$$