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We see that ${p}^{2}{c}^{2}\ll {E}_{0}^{2}$ so $K\ll {E}_{0}$ and we can use the nonrelativistic kinetic energy:
Kinetic energy of ideal gas in equilibrium at 300 K is:
We see that these energies are of the same order of magnitude.
Check Your Understanding Find the de Broglie wavelength and kinetic energy of a free electron that travels at a speed of 0.75 c .
$\lambda =1.417\phantom{\rule{0.2em}{0ex}}\text{pm;}$ $K=261.56\phantom{\rule{0.2em}{0ex}}\text{keV}$
Which type of radiation is most suitable for the observation of diffraction patterns on crystalline solids; radio waves, visible light, or X-rays? Explain.
X-rays, best resolving power
Speculate as to how the diffraction patterns of a typical crystal would be affected if $\text{\gamma}\text{-rays}$ were used instead of X-rays.
If an electron and a proton are traveling at the same speed, which one has the shorter de Broglie wavelength?
proton
If a particle is accelerating, how does this affect its de Broglie wavelength?
Why is the wave-like nature of matter not observed every day for macroscopic objects?
negligibly small de Broglie’s wavelengths
What is the wavelength of a neutron at rest? Explain.
Why does the setup of Davisson–Germer experiment need to be enclosed in a vacuum chamber? Discuss what result you expect when the chamber is not evacuated.
to avoid collisions with air molecules
At what velocity will an electron have a wavelength of 1.00 m?
What is the de Broglie wavelength of an electron travelling at a speed of $5.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}\text{m/s}$ ?
145.5 pm
What is the de Broglie wavelength of an electron that is accelerated from rest through a potential difference of 20 keV?
What is the de Broglie wavelength of a proton whose kinetic energy is 2.0 MeV? 10.0 MeV?
20 fm; 9 fm
What is the de Broglie wavelength of a 10-kg football player running at a speed of 8.0 m/s?
(a) What is the energy of an electron whose de Broglie wavelength is that of a photon of yellow light with wavelength 590 nm? (b) What is the de Broglie wavelength of an electron whose energy is that of the photon of yellow light?
a. 2.103 eV; b. 0.846 nm
The de Broglie wavelength of a neutron is 0.01 nm. What is the speed and energy of this neutron?
What is the wavelength of an electron that is moving at a 3% of the speed of light?
80.9 pm
At what velocity does a proton have a 6.0-fm wavelength (about the size of a nucleus)? Give your answer in units of c .
What is the velocity of a 0.400-kg billiard ball if its wavelength is 7.50 fm?
$2.21\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\text{\u2212}20}\text{m/s}$
Find the wavelength of a proton that is moving at 1.00% of the speed of light (when $\beta =0.01).$
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