A laboratory measures electron levels in nanostructures and
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 laboratory measures electron levels in nanostructures and, separately, the microwave rotational spectrum of a gas of rigid linear molecules. Use ℏ2/2me=0.0381 eV nm2 and ℏ=1.055×10−34 J s. All confining walls are infinitely high, the electron interacts with nothing else, and only orbital states are counted (spin is ignored). For the rotor the bond length is fixed and a photon changes l by one.
((a)) An electron is confined in a cubic box of side 2.0 nm. How many states have nx2+ny2+nz2=9, and what energy in eV do they share?
((b)) The same electron is instead confined in a hollow sphere of radius 2.0 nm, with V=0 inside and V infinite outside. Find the energy in eV of the lowest level, which has l=0.
((c)) A rigid rotor has moment of inertia I=2.00×10−46 kg m2. At what frequency, in GHz, does the l=0→l=1 line appear?