A silicon nanocrystal traps a single electron in a cubic box of side nm with…
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 silicon nanocrystal traps a single electron in a cubic box of side a=1.5 nm with infinitely high walls; the electron interacts with nothing else and spin is ignored throughout. Use ℏ2/2me=0.0381 eV nm2 and ℏ=1.055×10−34 J s. The same electron, modelled instead as trapped in a hollow sphere of the same radius (V=0 inside, infinite outside), has its lowest l=1 level at β=4.49, the first zero of the l=1 radial function.
((a)) List the three lowest values of nx2+ny2+nz2 for the cubic box, and give the degeneracy and the energy, in eV, of each of the three levels.
((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? Give the factor to three significant figures, and say which is the larger energy.
((c)) A rigid rotor's l=1→l=2 line is observed at 120.0 GHz. Find its moment of inertia, in kg m2, and the frequency of its lowest line, l=0→l=1, in GHz.