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

Options

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

Answer

B

Explanation

`p=h/lambdaRigtharrowv=h/(mlambda)and nlambda/2=lambda=(2a)/n`

`therefore=h/m n/(2a)Rightarrowvpropn`