Problem 6.92

The wave function of an electron of a hydrogen atom in the ground state takes the form \(\psi(r)=A \mathrm{e}^{-r / r_{1}},\) where \(A\) is a certain constant, \(r_{1}\) is the first Bohr radius. Find: (a) the most probable distance between the electron and the nucleus: (b) the mean value of modulus of the Coulomb force acting on the electron; (c) the mean value of the potential energy of the electron in the field of the nucleus.

6.92 .\) (a) The probability of the electron being at the interval \(r, r+d r\) from the nucleus is \(d P=\psi^{2}(r) 4 \pi r^{2} d r .\) From the condition for the maximum of the function dP/dr we get \(r_{p r}=r_{1} ;\) (b) \(\langle F\rangle=2 e^{2} / r_{1}^{2}\) (c) \(\langle U\rangle=-e^{2} / r_{1}

Wave Properties of particles. Schrodinger Equation.