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Compute the typical de Broglie wavelength of an electron in a metal at 27 ºC and compare it with the mean separation between two electrons in a metal which is given to be about 2 * 10^{-10} m.

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Anonym0us
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[Note: Exercises 11.35 and 11.36 reveal that while the wave-packets associated with gaseous molecules under ordinary conditions are non-overlapping, the electron wave-packets in a metal strongly overlap with one another. This suggests that whereas molecules in an ordinary gas can be distinguished apart, electrons in a metal cannot be distinguished apart from one another. This indistinguishibility has many fundamental implications which you will explore in more advanced Physics courses.]

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De Broglie wavelength (video) | Khan Academy

Bohr model energy levels (derivation using physics) ... Since visible light has a wavelength of about 500 nanometers, this means that visible light ... Since electrons have a rest mass, unlike photons, they have a de Broglie wavelength ... For example in the double slit experiment.... electron waves diffract and then ' condense' ...

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I found an answer from stanford.edu

PHYSICS 430 Lecture Notes on Quantum Mechanics

The wave equation for De Broglie waves, and the Born interpretation. Why an electron is not a wave. 4. The Quantum State. How does the electron get from A to ...

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I found an answer from physics.stackexchange.com

Kinetic energy of electron in metals - Physics Stack Exchange

Sep 2, 2016 ... At room temperature the electrons in metals are actually a degenerate Fermi gas and can be treated as if near absolute zero. Quoting wikipedia ...

For more information, see Kinetic energy of electron in metals - Physics Stack Exchange

Veda
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Given that

de Broglie wavelength of an electron \lambda = ?

Temperature T = 27\degree C

Mean separation between two electrons r = 2 * 10^{-10}

Step 1: Ge an expression for the de Broglie wavelength of an electron

Gaseous particle kinetic energy at a certain temperature  K.E = \frac{3}{2} kT ............................(1)

Where,  k - Boltzmann constant. = 1.38 * 10^{-23} J/K

de Broglie wavelength \lambda = \frac{h}{p}

Where, Plank constant h = 6.63 * 10^{-34} and p - momentum

\lambda = \frac{h}{\sqrt{2mK.E}}                                \because K.E = \frac{p^2}{2m}

\lambda = \frac{h}{3mkT}                                      \because \text{ equation (1)}

Step 2: Finding the de Broglie wavelength of an electron

Mass of an electron m = 9.11 * 10^{-31} kg

\lambda = \frac{6.63 * 10^{-34}}{3 * 9.11 * 10^{-31} * 1.38 * 10^{-23} * 300 }

= 6.2 * 10^{- 9} m

de Broglie wavelength of an electron \lambda=6.2*10^{-9} m

Therefore, Mean separation between two electrons < de Broglie wavelength of an electron (r < \lambda)