“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 — A five-atom gold wire

Exercises

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

Practice Problem 6 of 10

A five-atom gold wire

Problem

An STM builds a chain of 5 gold atoms, spacing 2.88 Å — a box of

L = 1.44

nm holding the 5 valence electrons (one 6s electron per atom). Where is the HOMO, and what wavelength does the chain absorb at?

Solution

Predict before reading on. This is the worked example wearing a lab coat: fill levels, difference the frontier pair, convert. One wrinkle — an odd electron count. Where does the fifth electron go?

Two electrons each in $n = 1, 2$ ; the fifth sits alone in $n = 3$ . The lowest absorption is still $3 \to 4$ :

Δ E = (16 - 9) \frac{h ^{2}}{8 m _{e} L ^{2}} = 1.27 eV \Rightarrow λ = 977 nm

Near-infrared. Atom-by-atom chains like this are real STM experiments, and free-electron boxes are the standard first model of their level spectra.

Result

HOMO: n = 3 (half-filled), λ \approx 977 nm