“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 — Why nuclear physics costs MeV

Exercises

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

Practice Problem 3 of 10

Why nuclear physics costs MeV

Problem

A neutron is confined to a nucleus-sized box,

L = 8

fm. Compute

E_{1}

and the spacing

E_{2} - E_{1}

. Compare with the eV scale of the dye molecule and explain the ratio with one sentence of scaling.

Solution

Predict before reading on. The analogue of the worked example’s $E_{1}$ step, with two substitutions: $m_{e} \to m_{n}$ (heavier by 1839) and $L$ smaller by a factor $\sim 1 0^{5}$ . Predict which effect wins before computing.

E_{1} = \frac{h ^{2}}{8 m _{n} L ^{2}} = 3.20 MeV, E_{2} - E_{1} = 3 E_{1} = 9.59 MeV

Confinement wins: $E \propto 1/ (m L^{2})$ , and $(1 0^{5})^{2}$ of box-shrinking crushes the factor 1839 of extra mass. The MeV scale of nuclear spectra is not exotic physics — it is the same formula as the dye molecule with a femtometer plugged in.

Result

E_{1} = 3.20 MeV, E_{2} - E_{1} = 9.59 MeV