Question

Consider the hydrogen atom to be a proton embedded in a cavity of radius $a_0$ (Bohr radius) whose charge is neutralized by the addition of an electron to the cavity in vacuum infinitely slowly. Estimate the average total energy of an electron in its ground state in a hydrogen atom as the work done in the above neutralization process. Also, if the magnitude of average KE is half the magnitude of average potential energy, find the average potential energy.

Answer 1

$W=-e^2/4\pi E_o{a}_o$

$T.E=P.E+K.E=e^2/4\pi E_o{a}_o$

$K.E=e^2/8\pi E_o{a}_o$

$T.E=P.E+K.E=\frac{-e^2}{4\pi E_o{a}_o}+\frac{e^2}{8\pi E_o{a}_o}$

or $T.E=\frac{-e^2}{8\pi E_o{a}_o}$

$P.E=2\times T.E=2\times \frac{-e^2}{8\pi E_o{a}_o}\Rightarrow \frac{-e^2}{8\pi E_o{a}_o}$