Dependence of Allowed Energy on Box Length in Quantum Well

Paragraph for Question Nos. 56 to 58

When a particle is restricted to move along $x$-axis between $x = 0$ and $x = a$, where $a$ is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends $x = 0$ and $x = a$. The wavelength of this standing wave is related to the linear momentum $p$ of the particle according to the de Broglie relation. The energy of the particle of mass $m$ is related to its linear momentum as $E = \frac{p^2}{2m}$. Thus, the energy of the particle can be denoted by a quantum number '$n$' taking values $1, 2, 3, \dots$ ($n = 1$, called the ground state) corresponding to the number of loops in the standing wave.

Use the model described above to answer the following three questions for a particle moving in the line $x = 0$ to $x = a$. Take $h = 6.6\times 10^{-34}\text{ Js}$ and $e = 1.6\times 10^{-19}\text{ C}$.