A nucleus confines its protons and neutrons to a region of size fm ( fm m)
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Origins of quantum theory · Origins of quantum theory
Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Matter waves and wave-particle duality
Problem
A nucleus confines its protons and neutrons to a region of size Δx=4.4 fm (1 fm =10−15 m). Use the estimating form of the uncertainty relation, Δp∼ℏ/Δx, with ℏ=1.055×10−34 J s, mp=1.673×10−27 kg, me=9.109×10−31 kg, c=3.00×108 m s−1, mec2=0.511 MeV and 1 eV =1.602×10−19 J.
((a)) Estimate the least kinetic energy of a proton confined to this region, in MeV, treating it non-relativistically.
((b)) Repeat the estimate of Δp for an electron confined to the same region, and use it in the non-relativistic formula K=(Δp)2/2me, again in MeV. Compare the result with the electron's own rest energy mec2, and explain why this formula cannot be trusted here.
((c)) Chapter 1 gives the electron's relativistic energy as E=p2c2+me2c4, which reduces to E≈pc when pc≫mec2. Using the same Δp as in (b), estimate the electron's kinetic energy in this extreme-relativistic limit, in MeV.
((d)) Beta decay ejects electrons from nuclei of this size with observed energies typically of a few MeV. Compare this with your answer to (c), and state in one sentence what the comparison suggests about whether an electron could exist, ready-formed, inside the nucleus before the decay.