When a particle is restricted to move along x- axis between `x = 0 and x = 4 ` whwre a is opf nanometer demension , its energy can take only certain spscfic values . The allowed energies of the particles only in such a restiricted regain , correspond to the formation of standing wave with nodes at its end ` x = 0 and x = a `.The wavelength of this standing wave is related to the linear momentum p of the paarticle according to the de Broglie relation .The energy of the particle of mass `m` is reated to its linear momentum as
`E = (p^(2))/(2m)` . thus , the energy of the particle can be denoted by a quantum number `n` taking value `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 there question 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 speed of the particle , that can take discrete values, is propotional to

Options

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

Answer

D

Explanation

lambda = (h)/(p) rArr lambda = (h)/(m nu) rArr m nu = (h)/(lambda)`
`But (n lambda)/(2) = a rArr lambda = (2a)/(n)`
`:. M nu = (nh)/(2a) rArr nu = (nh)/(2am) rArr nu prop n `