“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 — Swap the particle, rescale the spectrum

Exercises

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

Practice Problem 8 of 10

Swap the particle, rescale the spectrum

Problem

A muon is 207 times heavier than an electron. Without recomputing any integral, use the

L = 0.1

nm result (

E_{1} = 37.6

eV for an electron) to find

E_{1}

for a muon in the same box, and state the general scaling rule for

m

and

L

in one line.

Solution

Predict before reading on. Pure scaling — the analogue of the worked example is its formula, not its arithmetic. Which symbols does $E_{1}$ depend on, and with what powers?

E_{n} \propto \frac{1}{m L ^{2}} \Rightarrow E_{1}^{(μ)} = \frac{37.6 eV}{207} = 0.182 eV

One line of scaling replaces the whole computation — and it is the same line that predicted MeV physics from femtometer boxes two problems ago. Muonic atoms really do have their spectra compressed by the mass ratio (and their orbits shrunk by it), which is how muonic hydrogen measures the proton’s radius.

Result

E \propto 1/ (m L^{2}); E_{1}^{(μ)} = 0.182 eV