Physics 4307

Introduction to Quantum Mechanics Homework Assignment 5 Due Date: 2:00 p.m. on Thursday, November 12, 2015

1. Find all bound state energies for an electron in the symmetric finite well of V(x) = 0 for -L/2 ≤ x ≤ +L/2 with L = 2nm and V(x) = 13.6056923 eV = 1.0 Ryd = 0.5 Eh = 0.5 au elsewhere. Plot the four lowest stationary state solutions for the well. Compare the lowest four allowed energies for this well to those of the infinite square well of the same width.
2. For the previous finite well problem, what happens to the number of allowed states as the well becomes narrower (L < 2 nm)? Does the value of the ground state energy change? If so, how does the ground state energy of the narrower well differ from that of the 2 nm well?
3. What happens to the number of states and the value of the ground state energy as the original finite well becomes wider (L > 2nm)?
4. For the previous finite well problem with L = 2 nm, what happens to the ground state energy of the well as the walls become harder, i.e., as V(x) increases at +/- L/2, e.g., for V(x) >> 13.6 eV?
5. For the 1-D delta function well: V(x) = -α δ(x) where α is a positive constant and V(x) = 0 for x ≠ 0. Find the reflection (R) and transmission (T) coefficients for an E > 0 plane wave incident from the left.