Exercise:
The wave function for the ground state of a hydrogen atom is given by psi_r frace^-r/a_pi a_^/ abcliste abc Show that psi_ is normalised i.e. that the volume egral of its square modulus is equal to : _V |psi_r|^ r^ textrmdV abc The radial probability density for a radially symmetrical wave function is Pr pi r^ |psi_r|^ Calculate the radius with the highest probability density. abc Calculate the expectation value for the radius operator hat r which is defined through its action on the wave function: hat r psi_r r psi_r abc Calculate the uncertay for the radius operator. abcliste

Solution:
abcliste abc _V |psi_r| r^ textrmdV pi^_^infty |frace^-r/a_pi a_^/|^ r^ textrmdr fracpi^pi^ a_^ _^infty r^ lefte^-r/a_right textrmdr fraca_^frac a_^ quad square where we have used the general expression for the egral I'_n _^infty r^n lefte^-r/aright^ textrmdr fracn! a^n+^n+ see exercises for n. abc The radial probability density for the ground state is Pr pi r^ |psi_r|^ fracpi r^pi^ a_^ e^-r/a_ fraca_^ r^ e^-r/a_ This expression has extremal pos for P'r Longrightarrow fractextrmdtextrmdrleftr^ e^-r/a_right r e^-r/a_ - r^ e^-r/a_-/a_ e^-r/a_leftr-fracr^a_right Longrightarrow r r-a_ Longrightarrow r quad textrmor quad r a_ The solution r is a minimum psi_ but ra_ is the maximum. It follows that the Bohr radius a_ is the radius with the highest radial probability in the ground state. abc The expectation value for the radius operator is langle hat r rangle _V psi^*_r r psi_r r^ textrmdV pi^ _^infty |psi_r|^ r r^ textrmdr fracpi^pi^ a_^ _^infty e^-r/a_ r^ textrmdr fraca_^ frac a_^^ fraca_ The expectation value is / times the Bohr radius. abc The expectation value for the square of the radius operator can be calculated in the same way: langle hat r^ rangle _V psi^*_r r^ psi_r r^ textrmdV pi^ _^infty |psi_r|^ r^ r^ textrmdr fracpi^pi^ a_^ _^infty e^-r/a_ r^ textrmdr fraca_^ frac a_^ a_^ It follows for the uncertay standard deviation of the radius sigma sqrtlangle hat r^ rangllangle hat r rangle^ sqrt a_^-leftfrac a_right^ sqrtfraca_^ fracsqrt a_ abcliste The calculations can be verified or exted to other states using the Mathematica file linked to from this exercise.