Example III. Bound states of a Morse potential:

A deep enough potential well will support bound states. Even the attractive interaction of a He atom with a solid surface will support a few bound states. Here you will generate bound state wavefunctions for the Morse potential (corresponding to E<0) and determine the allowed bound state energy levels. The procedure is to guess what the bound state energy might be, generate a solution to the Schrödinger equation using the recursion method and then improve our guess.

Start the iterations as before `deep enough' inside the classically forbidden region on the left, choosing a value of the energy close to . As you reach the classically forbidden region on the right of the well, the wavefunction will either blow up or blow down, depending on whether the energy you choose is slightly above or below the true bound state energy (if the energy is substantially higher, your wavefunction will approximate an excited state rather than the ground state). Only if you happen to choose an energy value that is exactly one of the allowed bound state energy levels (and in the absence of numerical errors) will the wavefunction go to zero at large distances (as a true bound state wavefunction should). Here it is not appropriate to normalize the wavefunction by dividing by the maximum value. Since the magnitude of the wavefunction can be very large after it blows up or down, the normalization by the maximum value can make the wavefunction on the well region look flat. Change the normalization here and divide by the value of the wavefunction at the position of the potential minimum. Find how many steps takes you to that position, nstmin, and divide the wavefunction by wavefn[[nstmin]]. Choose and plot the wavefunction. Then redo the calculation with a slightly lower energy value and compare. Refine the energy value in an attempt to get closer to the true bound state energy. Compare the energy values you determine numerically for the Morse potential with the solution to the harmonic oscillator potential function that best approximates the Morse potential near the potential minimum (to good approximation the lowest vibrational states of a diatomic molecule can be represented by the harmonic oscillator bound states). Use a Taylor expansion to get the harmonic force constant. You can easily get a Taylor expansion of a function in Mathematica using the command

Series[f[x], {x, a, m}]

This gives a Taylor expansion of the function f(x) at the point x=a up to the power m.

How well do the numerically calculated bound states compare with the harmonic oscillator bound states? Is there a systematic difference? How do the wavefunctions compare? Give the program a substantially higher guess for the energy. After converging to a good value for the bound state energy, determine which energy level this wavefunction corresponds to, just from looking at the shape of the function. How would you normalize the numerically generated bound state wavefunction? The Sum command is very convenient here,

Sum[array[[i]]2,{i,1,n}]}

and can be used to calculate the sum of the elements squared in the array starting from 1 and going up to n.

More Advanced Projects: If you have time and energy, try one of the following exercises: