When a particle is restricted to move along x-axis between x=0 and x=a , where a is of nanometer dimesion , 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 node at its end 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=2mp_{2} $ . 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 $h=6.6×10_{−34}Js$ and $e=1.6×10_{−19}C$.

The speed of the particle , that can take discrete values, is proportional to