$A$ 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 ....... .

The ground state energy of a hydrogen atom is $-13.6 \text{ eV}$. What is the kinetic energy of the electron in this state (in $\text{ eV}$)?

What is the ratio of the wavelengths emitted during the $2 \to 1$ transition in $Li^{++}$,$He^+$,and $H$?

$A$ light of energy $12.75 \; eV$ is incident on a hydrogen atom in its ground state. The atom absorbs the radiation and reaches to one of its excited states. The angular momentum of the atom in the excited state is $\frac{x}{\pi} \times 10^{-17} \; eVs$. The value of $x$ is $........$ (use $h=4.14 \times 10^{-15} \; eVs$)

If an electron is moving in the $4^{\text{th}}$ orbit of the hydrogen atom,then the angular momentum of the electron in $SI$ units is

The speed of an electron in the orbit of a hydrogen atom in the ground state is