Question

When a particle is restricted to move along x-axis between `x=0` and `x=a`, where `alpha` if of nenometer 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.
Use the model described above to answer the following three questions for a particle moving along the line from `x=0` to `x=alpha`. Take `h=6.6xx10^(-34)Js` and `e=1.6xx10^(-19)` C.
Q. The allowed energy for the particle for a particular value of n is proportional to
A. `a^(-2)`
B. `a^(3//2)`
C. `a^(-1)`
D. `a^(2)`

Answer 1

Correct Answer - A

`n(lambda)/(2) = a` and `p = (h)/(lambda) = (nh)/(2a)`
Energy `E = (p^(2))/(2m) = (n^(2)h^(2))/(8 ma^(2)) rArr E prop (1)/(a^(2))`