Question

When a particle is restricted to move along the 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 $E=p^2/2m.$ Thus, the energy of the particle can be denoted by a quantum number 'n' taking values 1,2,3,.... (n=1, called the ground state) corresponding to the number of loops in the standing wave.
Take $h=6.6\times {10}^{-34}Jsande=1.6\times {10}^{-19}C.$

The allowed energy for the particle for a particular value of n is proportional to

Answer 1

$a=\frac{n\lambda }{2}\Rightarrow \lambda =\frac{2a}{n}$
${\lambda }_{deBroglie}=\frac{h}{p}$
$\frac{2a}{n}=\frac{h}{p}\Rightarrow p=\frac{nh}{2a}$
$E=\frac{p^2}{2m}=\frac{n^2{h}^2}{8a^{2}m}$
$\Rightarrow E\alpha l/a^2$