In a Bohr atom,the total energy of an electron in the $n$-th allowed orbit is $E_n$ and its angular momentum is $J_n$. Then:

In the Bohr model of the hydrogen atom,the centripetal force is provided by the Coulomb attraction between the proton and the electron. If $r_0$ is the radius of the ground state orbit,$m$ is the mass,$e$ is the charge on the electron,and $\varepsilon_0$ is the permittivity of vacuum,the speed of the electron is:

Excitation energy of a hydrogen-like ion in its first excitation state is $40.8 \, eV$. Energy needed to remove the electron from the ion in the ground state is ........ $eV$.

$A$ hydrogen atom is in its $n^{\text{th}}$ energy state. If the de-Broglie wavelength of the electron is $\lambda$,then:

In a hydrogen atom,an electron makes a transition from the $5^{\text{th}}$ orbit to the $3^{\text{rd}}$ orbit. The change in the angular momentum for this electron is . . . . . . .

Four electrons,each of mass $m_{e}$,are in a one-dimensional box of size $L$. Assume that the electrons are non-interacting,obey the Pauli exclusion principle,and are described by standing de Broglie waves confined within the box. Define $\alpha = h^{2} / 8 m_{e} L^{2}$ and $U_{0}$ to be the ground state energy. Then,