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Atoms and Nuclei question

2009 · Shift 1 · Q55
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Atoms and Nuclei question

2009 · Shift 1 · Q55

JEE AdvancedPhysicsAtoms and NucleiMCQ+3 / −1
When a particle is restricted to move along 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 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=p22mE = {{{p^2}} \over {2m}}E=2mp2​. 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 in the line x = 0 to x = a. Take h=6.6×10−34h = 6.6 \times {10^{ - 34}}h=6.6×10−34 J-s and e=1.6×10−19e = 1.6 \times {10^{ - 19}}e=1.6×10−19 C.The speed of the particle, that can take discrete values, is proportional to
  1. A
    n−3/2{n^{ - 3/2}}n−3/2
  2. B
    n−1{n^{ - 1}}n−1
  3. C
    n1/2{n^{1/2}}n1/2
  4. D
    nnn
View written solutionFree

Correct answer: D

  1. Standing wave condition

For a particle confined between x=0x=0x=0 and x=ax=ax=a, the wave must have nodes at both ends. Hence the allowed wavelengths satisfy

a=nλn2,n=1,2,3,…a = n\frac{\lambda_n}{2}, \qquad n=1,2,3,\dotsa=n2λn​​,n=1,2,3,…

So,

λn=2an\lambda_n = \frac{2a}{n}λn​=n2a​

  1. Use de Broglie relation

The de Broglie relation gives

pn=hλnp_n = \frac{h}{\lambda_n}pn​=λn​h​

Substituting λn=2an\lambda_n = \dfrac{2a}{n}λn​=n2a​,

pn=h2a/n=nh2ap_n = \frac{h}{2a/n} = \frac{nh}{2a}pn​=2a/nh​=2anh​

Thus,

pn∝np_n \propto npn​∝n

  1. Relate speed to momentum

Since linear momentum is

pn=mvnp_n = mv_npn​=mvn​

for a particle of fixed mass mmm,

vn=pnm=nh2amv_n = \frac{p_n}{m} = \frac{nh}{2am}vn​=mpn​​=2amnh​

Hence,

vn∝nv_n \propto nvn​∝n

  1. Check options
  • A: n−3/2n^{-3/2}n−3/2 ❌
  • B: n−1n^{-1}n−1 ❌
  • C: n1/2n^{1/2}n1/2 ❌
  • D: nnn ✅

Therefore, the speed is proportional to nnn.

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