Introduction to Quantum Mechanics · Problem 1.18

The Wave Function — David J. Griffiths

Physics — Quantum Mechanics · de_broglie_wavelength_classical_limit

Problem

Very roughly speaking, quantum mechanics is relevant when the de Broglie wavelength of the particle in question (h/ph/p) is greater than the characteristic size of the system (dd). In thermal equilibrium at (Kelvin) temperature TT, the average kinetic energy of a particle is p22m=32kBT\frac{p^2}{2m} = \frac{3}{2}k_B T (where kBk_B is Boltzmann's constant), so the typical de Broglie wavelength is λ=h3mkBT.(1.45)\lambda = \frac{h}{\sqrt{3mk_BT}}.\tag{1.45} 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=0.3d = 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 PP quantum mechanical? *Hint:* Use the ideal gas law (PV=NkBTPV = Nk_BT) to deduce the interatomic spacing. *Answer:* T<(1/kB)(h2/3m)3/5P2/5T < (1/k_B)\left(h^2/3m\right)^{3/5} P^{2/5}. Obviously (for the gas to show quantum behavior) we want mm to be as *small* as possible, and PP 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 H2_2.)
Intuitive explanation

The key insight is comparing two length scales: the de Broglie wavelength λ\lambda (how "spread out" the particle is quantum mechanically) versus the system size dd (how much room it has to move). When λ>d\lambda > d, the particle's quantum fuzziness matters—it can't be confined to a definite position, so classical mechanics fails. You'll use the thermal de Broglie wavelength formula to find the temperature threshold where quantum effects kick in, then apply it to different systems by estimating their characteristic sizes.

Intuitive explanation

The key insight is comparing two length scales: the de Broglie wavelength λ\lambda (how "spread out" the particle is quantum mechanically) versus the system size dd (how much room it has to move). When λ>d\lambda > d, the particle's quantum fuzziness matters—it can't be confined to a definite position, so classical mechanics fails. You'll use the thermal de Broglie wavelength formula to find the temperature threshold where quantum effects kick in, then apply it to different systems by estimating their characteristic sizes.

Related theory

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