Consider electron in the ground state of the hydrogen atom, whose wavefunction takes

the simple form

(F ) =

1

nose r/a

(7)

where a is the Bohr radius.

First, show this state is properly normalized.

Given this state, we can define a probability density function along the r direc-

tion as

(8)

such that [ p(r)dr = 1. Find the radial position r at which p(r) is maximized, in

unit of the Bobr radius.

Derive the formula for the average of Coulomb potential energy (operator)

V(r) =

AREor

(9)

under the ground state, where co is the vacuum permittivity. In this result, replace the

Bohr radius a with 4xEgh?/(me?), where / is the reduced mass of the electron-proton

system.

Numerically, we know the ground state energy of hydrogen atom is -13.6 eV.

What is the value of this average potential energy?