Question

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.

[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.]

Answer 1

Temperature, T = 27°C = 27 + 273 = 300 K

Mean separation between two electrons, r = 2 × 10⁻¹⁰ m

De Broglie wavelength of an electron is given as:

Where,

h = Planck’s constant = 6.6 × 10⁻³⁴ Js

m = Mass of an electron = 9.11 × 10⁻³¹ kg

k = Boltzmann constant = 1.38 × 10⁻²³ J mol⁻¹ K⁻¹

Hence, the de Broglie wavelength is much greater than the given inter-electron separation.