Ratio of centripetal acceleration for an electron revolving in $3^{\text{rd}}$ and $5^{\text{th}}$ Bohr orbit of hydrogen atom is

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:

Write the formula for the orbital radius of the electron in the atom based on the Bohr atomic model.

The radius of the first orbit in an $H$-atom is '$a_0$'. Then,the de-Broglie wavelength of the electron in the third orbit is: (in $\pi a_0$)

The gravitational force between a $H$-atom and another particle of mass $m$ is given by Newton's law $F = G\frac{M m}{r^2}$. Here,$M$ represents:

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,