In general, quantum mechanics is relevant when the de Broglie wavelength of the particle i...

In general, 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

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

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The purpose of this problem is to anticipate 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 free18 electrons in a solid are quantum mechanical. Below what temperature are the nuclei in a solid quantum mechanical? (Use sodium as a typical case.) Moral: The free electrons in a solid are always quantum mechanical; the nuclei are almost never 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 = NkBT) to deduce the interatomic spacing. Answer: T<(1/kB)(h2/3m)3/5 P2/5 Obviously (for the gas to show quantum behavior) we wantm 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?