When a particle is restricted to move ao...

When a particle is restricted to move aong `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 realated to the linear momentum `p` of the particle according to the de Breogile relation. The energy of the particl e of mass m is reelated 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 loop in the standing wave.
Use the model decribed above to answer the following three questions for a particle moving in the line `x = 0` to `x =a`. Take `h = 6.6 xx 10^(-34) J s` and `e = 1.6 xx 10^(-19) C`.
The allowed energy for the particle for a particular value of n is proportional to

`a^(-2)`

`a^(3//2)`

`a^(-1)`

`a^(2)`

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