Question

Expectation of radii of hydrogen atom wave function derivation

Answer 1

The average or "expectation value" of the radius for the electron in the ground state of hydrogen is obtained from the integral

This requires integration by parts. The solution is

All the terms containing r are zero, leaving

It may seem a bit surprising that the average value of r is 1.5 x the first Bohr radius, which is the most probable value. The extended tail of the probability density accounts for the average being greater than the most probable value.