A crude model confines a single neutron in a cubic cavity of side fm with infinitely…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Central potentials and canonical three-dimensional systems · Central potentials and canonical three-dimensional systems
Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The Schrödinger equation in three dimensions and central potentials
Problem
A crude model confines a single neutron in a cubic cavity of side a=6 fm with infinitely high walls, the neutron interacting with nothing else; spin is ignored in the counting. Use ℏ2/2mn=20.7 MeV fm2 and ℏ=1.055×10−34 J s. The same neutron, modelled instead as trapped in a hollow sphere of the same radius, has its lowest l=1 level at β=4.49, the first zero of the l=1 radial function.
((a)) A level of the cubic cavity is six-fold degenerate only when nx2+ny2+nz2 comes from three different positive integers. Find the two lowest values of nx2+ny2+nz2 built from three different integers, and give the energy of each level, in MeV.
((b)) For the hollow sphere of the same radius, by what factor does the energy of the lowest l=1 level exceed that of the lowest l=0 level?
((c)) A rigid rotor's rotational lines are equally spaced by 1.30 THz. Find its moment of inertia, in kg m2, and the wavelength of its lowest line, in mm.