1. Consider a particle of mass subject to a one-dimensional potential of the following form:
V(x) = {1/2kx2 for x>0, +∞ for x<0
This is a combination of the particle in a box and the harmonic oscillator that might be a better model for real diatomics than the standard harmonic oscillator. On the right side of x=0, the Hamiltonian is exactly the same as a harmonic oscillator Hamiltonian. The hard wall at x=0, however, introduces a boundary condition. Use what you have learned about both the ordinary harmonic oscillator and the particle-in-a-box boundary conditions to answer the following questions:
a) What does this boundary condition require the wave functions to do at x=0?
b) Which of the normal harmonic oscillator wavefunctions have the property required in part (a) and would then be valid functions in the new Hamiltonian?
c) If the lowest of these "allowed" states is the ground state, what is the ground state energy of this system?
d) What would the energy of the first excited state be?
e) What is the spacing in energy between any two adjacent levels? (Hint: You don't need to solve any integrals to do this problem. You do, however, need to think a little more deeply about this one than you would for a standard plug-and-chug problem.)