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De Broglie & Uncertainty Q

This document covers key concepts in quantum mechanics including de Broglie waves, the uncertainty principle, and their applications. It contains 16 questions that define phase and group velocity of waves, calculate de Broglie wavelengths, relate momentum and wavelength using Bohr's model of the hydrogen atom, derive expressions for relativistic de Broglie wavelength, distinguish phase and group velocity, explain Heisenberg's uncertainty principle and how it prevents electrons from existing within the nucleus.

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Bishnu gopal Das
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64 views1 page

De Broglie & Uncertainty Q

This document covers key concepts in quantum mechanics including de Broglie waves, the uncertainty principle, and their applications. It contains 16 questions that define phase and group velocity of waves, calculate de Broglie wavelengths, relate momentum and wavelength using Bohr's model of the hydrogen atom, derive expressions for relativistic de Broglie wavelength, distinguish phase and group velocity, explain Heisenberg's uncertainty principle and how it prevents electrons from existing within the nucleus.

Uploaded by

Bishnu gopal Das
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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De Broglie & Uncertainty

1. Define phase velocity of a wave and show that a particle in motion cannot be represented by a
single wave. Hence define group velocity. 1+2+2
2. Calculate the de Broglie wavelength of a baseball of mass 1 kg moving at a speed of 10 m/s.
Discuss the reason why its wave nature cannot be observed. 3+2
3. A free electron has de Broglie wavelength 5 pm. Find the phase and group velocities of its de
Broglie wave. 3+2
4. What do you mean by matter waves? State and explain de Broglie’s hypothesis of matter waves.
Write down the characteristics of this type of waves. 1+2+2
5. If 𝑟𝑛 be the radius of the 𝑛th Bohr orbit such that 𝑛 number of waves with wavelength 𝜆 may be
fitted into the orbit, then find out the ratio of the Planck’s constant to the momentum using Bohr’s
quantum condition. What is the significance of this equation? 4+1
6. If an electron is subjected to a potential difference of 𝑉 volt, then show that the corresponding de
Broglie wavelength is 𝜆 = 12.26⁄√𝑉 Å. 4
7. If a particle of charge 𝑒 and rest mass 𝑚0 is accelerated by a potential 𝑉 then find the de Broglie
wavelength of the particle viewing the relativistic energy momentum relation. 4
8. State de Broglie hypothesis. Show that the relativistic de Broglie wavelength is given by
ℎ𝑐
𝜆= , Where the notations used have their usual meanings. 4
√𝐸𝑘 (𝐸𝑘 +2𝑚0 𝑐 2 )
9. Distinguish between phase velocity and group velocity. Show that the relation between group
d vp
vg = v p − 
velocity (vg) and phase velocity (vp) is given by d . 2+4
10. Prove that the product of phase velocity and group velocity for a de Broglie wave is constant for
relativistic cases. 3
11. Show that the group velocity of the wave packet representing a particle is equal to the velocity of
the particle itself. 3
12. State and explain Heisenberg’s uncertainty principle regarding the canonically conjugate pair of
position and momentum. 3
13. Explain the non-existence of electrons within the nucleus of an atom from the Heisenberg’s
uncertainty relation. 5
14. To what velocity must we accelerate a particle so that its Compton wavelength equals to its de
Broglie wavelength? 3
15. Compute the smallest possible uncertainty in the position of an electron moving with velocity
3×107 m/s. The rest mass of electron is 9.1×10-31 kg. 3
16. The maximum uncertainty in the position of an electron in a nucleus is 2×10-14 m. Find the
minimum momentum, given h = 6.626×10-34 Js. 3

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