The angular momentum of an electron in a hydrogen atom is proportional to: (Where $r$ is the radius of the orbit of the electron)

$A$ stationary hydrogen atom undergoes a transition from $n=5$ to $n=4$. The recoil speed of the atom is (where $R=$ Rydberg constant,$h=$ Planck's constant,and $m=$ mass of the hydrogen atom).

$A$ hydrogen atom,a deuteron atom,a $He^+$ ion,and a $Li^{++}$ ion each contain a single electron orbiting their nucleus. The wavelengths of the electromagnetic radiation emitted during the transition of the electron from the $n = 2$ orbit to the $n = 1$ orbit are found to be $\lambda_1, \lambda_2, \lambda_3$,and $\lambda_4$ respectively. Then:

In a hydrogen atom,find the magnetic field at the center in the ground state. Given that Bohr's radius is $r_{0} = 5 \times 10^{-11} \, m$ (in $T$).

$P$ is the probability of finding the $1s$ electron of a hydrogen atom in a spherical shell of infinitesimal thickness, $dr$, at a distance $r$ from the nucleus. The volume of this shell is $4\pi r^2 dr$. The qualitative sketch of the dependence of $P$ on $r$ is:

The potential energy of a proton and an electron in a hydrogen atom is given by $V = V_0 \ln(r/r_0)$,where $r_0$ is a constant. Assuming the Bohr model is applicable to this system,find the relationship between the radius $r_n$ and the principal quantum number $n$.