A quantum dot holding one electron is modelled in two ways
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
A quantum dot holding one electron is modelled in two ways. Model A is a cubic box of side a=1.2 nm with impenetrable walls (V=0 inside, infinite outside). Model B is an isotropic three-dimensional harmonic oscillator, V=21meω2(x2+y2+z2), whose ℏω is chosen so that the two models have the same ground-state energy. Ignore spin throughout, and treat the electron as free of all other interactions.
((a)) Find the ground-state energy of model A, in eV.
((b)) Find the energy of the first excited level of model A, in eV, and the number of states it contains.
((c)) Find the value of ℏω for model B, in eV.
((d)) Find the energy of the first excited shell of model B, in eV, and the number of states it contains.
((e)) Every state of that first excited shell of the oscillator has l=1. Give the eigenvalue of L^2 for these states as a multiple of ℏ2.Take ℏ2/2me=0.0381 eV nm2.