Compute the typical de Broglie wavelength of an electron in a metal at 27 °C and compare it with the mean separation between two electrons in a metal which is given to be about 2 × 10–10 m.
[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 indistinguishability has many fundamental implications which you will explore in more advanced Physics courses.]
Given:
Temperature, T = 27° C = 300 K
Mean separation between electrons, r = 2 × 10-10 m
De-Broglie wavelength of electron is given by,
…(1)
Where,
h = Planck’s constant = 6.6 × 10-34 Js
m = mass of electron = 9.1 × 10-31 Kg
k = Boltzmann constant = 1.38 × 10-23 Jmol-1K-1
T = absolute temperature
Putting the values in equation(1), we get,
λ = 6.2 × 10-9 m
The De-Broglie wavelength is much larger than the interelectronic separation.