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 and the uncertay for the distance of the electron to the nucleus in the ground state. The distance operator hat r is defined through its action on the wave function: hat r psi_r r psi_r 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 distance 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. vspacemm The expectation value for the square of the distance 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 distance 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.