“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

The Particle in a Box by Hand — Zero-point energy vs room temperature

Exercises

Introductory Quantum Mechanics › Unit 2 · Bound states in one dimension › The Particle in a Box by Hand › all problems

Practice Problem 4 of 10

Zero-point energy vs room temperature

Problem

An electron is localized to atomic size,

L = 0.1

nm. Compare its ground-state energy with

k_{B} T

at 300 K. What does the ratio say about whether thermal agitation matters to atomic electrons?

Solution

Predict before reading on. One $E_{1}$ evaluation, then a comparison. The worked example compared two quantum levels; here the analogue of “the transition” is quantum-vs-thermal. Guess the order of magnitude of the ratio first.

E_{1} = \frac{h ^{2}}{8 m _{e} L ^{2}} = 37.6 eV, k_{B} T_{300 K} = 25.9 meV

Ratio $\approx 1455$ . An atomic electron’s zero-point energy dwarfs room-temperature thermal energy — which is why atoms do not “thermally unravel,” why electronic spectra are sharp at 300 K, and why chemistry at room temperature is quantum mechanics, not thermodynamics.

Result

E_{1} = 37.6 eV \approx 1455 \times k_{B} T_{300}