Home
Class 12
PHYSICS
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

Verified by Experts

The correct Answer is:
A
`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)`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • SIMPLE HARMONIC MOTION

    ALLEN|Exercise Comprehension #4|3 Videos

  • SIMPLE HARMONIC MOTION

    ALLEN|Exercise Comprehension #5|2 Videos

  • SIMPLE HARMONIC MOTION

    ALLEN|Exercise Comprehension #2|3 Videos

  • RACE

    ALLEN|Exercise Basic Maths (Wave Motion & Dopplers Effect) (Stationary waves & doppler effect, beats)|24 Videos

  • TEST PAPER

    ALLEN|Exercise PHYSICS|4 Videos

Similar Questions

Explore conceptually related problems

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

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. If the mass of the particle is m=1.0xx10^(-30) kg and alpha=6.6nm , the energy of the particle in its ground state is closest to

If the linear momentum of a prticle is known can you find its kinetic energy? If the kinetic energy of a particle is known can you find its linear momentum?

When the linear momentum of a particle is increased by 1% its kinetic energy increases by x%. When the kinetic energy of the particle is increased by 300%, its linear momentum increases by y%. The ratio of y to x is

What is the relation between wavelength and momentum of moving particles ?

For a particle moving along x- axis, speed must be increasing for the following graph :

Derive the relation between the wavelength (lambda) of the de broglie wave and kinetic energy (E ) of a moving particle

For a particle moving along the x-axis, mark the correct statement(s).

A particle with total energy E is moving in a potential energy region U(x). Motion of the particle is restricted to the region when

A particle moves along the x-axis from x=0 to x=5m under the influence of a given by F =7-2x + 3x^(2) . The work done by the applied force to the particle is.

Home
Profile