Assume that there is no repulsive force between the electrons in an atom but the force between positive and negative charges given by Coulomb’s law as usual. Under such circumstances, calculate the ground state energy of a He-atom.

Show that the first few frequencies of light that is emitted when electrons fall to the $n^{th}$ level from levels higher than $n$ are approximate harmonics (i.e. in the ratio $1:2:3$) when $n>>1$.

If a proton had a radius $R$ and the charge was uniformly distributed, calculate using Bohr theory, the ground state energy of a $\mathrm{H}$-atom when $R=0.1\mathrm{\AA }$

If a proton had a radius $R$ and the charge was uniformly distributed, calculate using Bohr theory, the ground state energy of a $\mathrm{H}$-atom when $R=10\mathrm{\AA }$

Use, Ground state radius $r_A=\frac{h^2}{4{\pi }^{\mathit{2}}mK{e}^{\mathit{2}}}=0.53\mathrm{\AA }$ , $P.E.=\frac{e^{\mathit{2}}}{\mathit{4}\pi {\epsilon }_{\mathit{0}}{r}_A}=27.2eV$

In the Auger process an atom makes a transition to a lower state without emitting a photon. The excess energy is transferred to an outer electron which may be ejected by the atom. (This is called an Auger electron). Assuming the nucleus to be massive, calculate the kinetic energy of an $\mathrm{n}=4$ Auger electron emitted by Chromium by absorbing the energy from a $\mathrm{n}=2$ to $\mathrm{n}=1$ transition.