An electron is bound in a finite square well for and outside
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Wave mechanics in one dimension
Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Stationary states and bound states in one dimension
Problem
An electron is bound in a finite square well V=−V0 for −a<x<a and V=0 outside. Energies are quoted above the floor of the well as E+V0, with ℏk=2me(E+V0), z=ka and z0=a2meV0/ℏ. The bound states are the crossings of tanz=(z0/z)2−1 (even) and −cotz=(z0/z)2−1 (odd), and in the deep limit z0→∞ the energies above the floor become those of an infinite square well of full width 2a; for any finite depth the ground state lies below that limiting value. Use ℏ2/2me=0.0381 eV nm2.
((a)) Well A has a=0.50 nm and V0=1.00 eV. Evaluate z0 and state how many bound states it holds.
((b)) Well B has the same depth but is three times as wide: a=1.50 nm, V0=1.00 eV. Evaluate z0 and state how many bound states it holds.
((c)) Evaluate, for each well, the deep-limit value of the ground-state energy above the floor, which is the upper bound just described, and state which way widening the well at fixed depth moves the ground state above the floor.