An electron is trapped between rigid walls at and nm
PHYSICS · Subjective · 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 trapped between rigid walls at x=0 and x=1.0 nm. The floor of the box has a step in it: V=0 for 0<x<0.5 nm and V=0.20 eV for 0.5nm<x<1.0 nm. All three of the lowest states have E>0.20 eV, so each of them oscillates in both halves of the box. A student sketches the three lowest states and makes four claims: (a) The three lowest states have 0, 1 and 2 nodes strictly between the walls. (b) In the raised half of the floor, where E−V is smaller, the wave function has a longer wavelength than in the deep half. (c) The state with two nodes is the ground state, because more nodes means a shorter wavelength, and a shorter wavelength means the electron is more tightly bound, so its energy is lowest. (d) None of the three states is even or odd about the midpoint x=0.5 nm, because the potential is not symmetric about that point. Exactly one claim is wrong. Say which, explain what goes wrong in the reasoning, and give the correct ordering of the three states by energy.