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Wave Properties of particles. Schrodinger Equation.

Problem 6.92

The wave function of an electron of a hydrogen atom in the ground state takes the form ψ(r)=Ae−r/r1,\psi(r)=A \mathrm{e}^{-r / r_{1}}, where AA is a certain constant, r1r_{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.

Reveal Answer
6.92.6.92 . (a) The probability of the electron being at the interval r,r+drr, r+d r from the nucleus is dP=ψ2(r)4πr2dr.d P=\psi^{2}(r) 4 \pi r^{2} d r . From the condition for the maximum of the function dP/dr we get rpr=r1;r_{p r}=r_{1} ; (b) ⟨F⟩=2e2/r12\langle F\rangle=2 e^{2} / r_{1}^{2} (c) ⟨U⟩=−e2/r1\langle U\rangle=-e^{2} / r_{1}
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