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When a particle is restricted to move al...

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

`alpha^(-2)`

B

`alpha^((-3)/(2))`

C

`alpha^(-1)`

D

`alpha^2`

Text Solution

Verified by Experts

`a=(nlamda)/(2)implieslamda=(2a)/(n)`
`lamda_(de Brogl ie)=(h)/(p)`
`(2a)/(n)=(h)/(p)impliesp=(nh)/(2a)`
`E=(p^2)/(2m)=(n^2h^2)/(8a^2m)`
`impliesEprop(1)/(a^2)`
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