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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

`a^(-2)`

B

`a^(-3//2)`

C

`a^(-1)`

D

`a^(2)`

Text Solution

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The correct Answer is:
A
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The relation between kinetic energy K and linear momentum p of a particle is represented by

Knowledge Check

  • 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

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

  • 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 alloewd 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)`

  • 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 speed of the particle, that can take disrete values, is proportional to

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