$A$ particle of mass $m$ moves in a circular orbit in a central potential field $U(r) = U_{0}r^{4}$. If Bohr's quantization conditions are applied,the radii of possible orbitals $r_{n}$ vary with $n^{1/\alpha}$,where $\alpha$ is ....... .

Find the ratio of the area of the orbit of the first excited state to the ground state in a hydrogen atom.

Experimentally it is found that $12.8 \, eV$ energy is required to separate a hydrogen atom into a proton and an electron. The orbital radius of the electron in this hydrogen atom is $\frac{9}{x} \times 10^{-10} \, m$. The value of $x$ is (Given: $1 \, eV = 1.6 \times 10^{-19} \, J$,$\frac{1}{4 \pi \epsilon_0} = 9 \times 10^9 \, Nm^2/C^2$,and electronic charge $e = 1.6 \times 10^{-19} \, C$)

An electron in Bohr's hydrogen atom has an energy of $-3.4 \ eV$. The angular momentum of the electron is

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 $n = 4$ Auger electron emitted by Chromium by absorbing the energy from an $n = 2$ to $n = 1$ transition.

The ratio of the speed of the electron in the $3^{rd}$ orbit of $He^{+}$ to the speed of the electron in the $3^{rd}$ orbit of the hydrogen atom is: