$A$ 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 ....... .

$A$ muon is an unstable particle with a mass of $207 \, m_e$ and a charge of either $+e$ or $-e$. $A$ muon $(\mu^-)$ is captured by a hydrogen nucleus to form a muonic atom. If the proton captures the $\mu^-$, find the ionization energy of this atom in $keV$.

The radius of the first orbit of hydrogen is $r_{H}$,and the energy in the ground state is $-13.6 \text{ eV}$. Considering a $\mu^{-}$-particle with a mass $207 m_e$ revolving around a proton as in a hydrogen atom,the energy and radius of the proton and $\mu^{-}$-combination respectively in the first orbit are (assume the nucleus to be stationary):

An excited hydrogen atom emits a photon of wavelength $\lambda$ in returning to the ground state. The quantum number $n$ of the excited state is ($R=$ Rydberg's constant).

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

The radius of the innermost electron orbit of a hydrogen atom is $5.3 \times 10^{-11} \;m$. What are the radii of the $n=2$ and $n=3$ orbits?