An electron is confined in a cubic box of side nm with impenetrable walls: the…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · 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
An electron is confined in a cubic box of side a=0.80 nm with impenetrable walls: the potential energy is zero inside and infinite outside, and the electron interacts with nothing else. Spin is ignored throughout, so each set of quantum numbers is one state. Take ℏ2/2me=0.0381 eV nm2.
((a)) Separating the box into three one-dimensional infinite wells, find the energy of the ground level and state its degeneracy.
((b)) Find the energy and the degeneracy of the level with nx2+ny2+nz2=6, and say how many interior nodal planes one of its eigenfunctions has.
((c)) The box is now replaced by an isotropic three-dimensional harmonic trap with ℏω=0.20 eV. Find the energy of the shell N=nx+ny+nz=3 and the number of states it contains.
((d)) A third confinement has a potential energy that depends only on r. For one solution of the radial equation with l=3, how many states share that radial function and its energy, and what is the eigenvalue of L^2 for their angular factor, in units of ℏ2?