Problem 1.18 – Griffith’s Intro to QM

Problem 1.18

Very roughly speaking, quantum mechanics is relevant when the de Broglie wavelength of the particle in question $(h/p)$ is greater than the characteristic size of the system $(d)$ . In thermal equilibrium at (Kelvin) temperature T, the average kinetic energy of a particle is

$\dfrac{p^2}{2m}=\dfrac{3}{2}k_BT$

(where $k_B$ is Boltzmann’s constant), so the typical de Broglie wavelength is

$\lambda = \dfrac{h}{\sqrt{3mk_BT}}$

The purpose of this problem is to determine which systems will have to be treated quantum mechanically, and which can safely be described classically.
(a) Solids. The lattice spacing in a typical solid is around $d=0.3$ nm. Find the temperature below which the unbound electrons in a solid are quantum mechanical. Below what temperature are the nuclei in a solid quantum mechanical? (Use silicon as an example.) Moral: The free electrons in a solid are always quantum mechanical; the nuclei are generally not quantum mechanical. The same goes for liquids (for which the interatomic spacing is roughly the same), with the exception of helium below 4 K.

(b) Gases. For what temperatures are the atoms in an ideal gas at pressure P quantum mechanical? Hint: Use the ideal gas law $(PV=Nk_BT)$ to deduce the interatomic spacing.

Answer: $T<(1/k_B)(h^2/3m)^{3/5}P^{2/5}$ . Obviously (for the gas to show quantum behavior) we want m to be as small as possible, and P as large as possible. Put in the numbers for helium at atmospheric pressure. Is hydrogen in outer space (where the interatomic spacing is about 1 cm and the temperature is 3 K) quantum mechanical? (Assume it’s monatomic hydrogen, not $H_2$ .)