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 

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? 

= = 3. KBT 2m (where kg is Boltzmann's constant), so the typical de Broglie wavelength is h 3mk BT (1.45)