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=p2/2m.
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.
The speed of the particle that a particle can take in discrete values is proportional to ( 'n' represents 'nth' orbit)