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

Options

A. `a^(-2)`
B. `a^(-3//2)`
C. `a^(-1)`
D. `a^(2)`

Answer

A

Explanation

`E=p^(2)/(2m)` …(i) `" "p=h/lambda` …(ii)
By equation (i) and (ii)
`E=h^(2)/(2mlambda^(2))implies (h^(2)(n^(2)))/(2m(4a^(2)))`
`[ :' (n lambda)/2=a` for stationary wave on string fixed at both end`] E prop a^(-2)`