The ratio between the total acceleration of the electron in a singly ionized helium atom and a hydrogen atom (both in the ground state) 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,

In the lowest energy level of a hydrogen atom,the electron has an angular momentum of:

Let $v_{n}$ and $E_{n}$ be the respective speed and energy of an electron in the $n$th orbit of radius $r_{n}$, in a hydrogen atom, as predicted by Bohr's model. Then:

$(a)$ Using the Bohr model,calculate the speed of the electron in a hydrogen atom in the $n = 1, 2,$ and $3$ levels.
$(b)$ Calculate the orbital period in each of these levels.

$A$ particle of mass $m$ is moving in a circular orbit under the influence of the central force $F(r) = -kr$,corresponding to the potential energy $V(r) = \frac{1}{2}kr^2$,where $k$ is a positive force constant and $r$ is the radial distance from the origin. According to Bohr's quantization rule,the angular momentum of the particle is given by $L = n\hbar$,where $\hbar = \frac{h}{2\pi}$,$h$ is Planck's constant,and $n$ is a positive integer. If $v$ and $E$ are the speed and total energy of the particle,respectively,then which of the following expression$(s)$ is(are) correct?
$(A)$ $r^2 = n\hbar \sqrt{\frac{1}{mk}}$
$(B)$ $v^2 = n\hbar \sqrt{\frac{k}{m^3}}$
$(C)$ $\frac{L}{mr^2} = \sqrt{\frac{k}{m}}$
$(D)$ $E = \frac{n\hbar}{2} \sqrt{\frac{k}{m}}$