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, . . . . . (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 $$h = 6.6 \times 10^{-34}$$J-s and $$e = 1.6 \times 10^{-19}$$ C.