In class, Prof. Miller showed that the energy levels for different potentials are spaced differently, and that the spacing is related to the shape of the potential. Specifically, he showed that the particle in a box energy levels become more widely spaced as the the energy level n gets higher, that the harmonic oscillator energy levels are always evenly spaced, and that the hydrogen atom energy levels become more closely spaced as n increases.

The Morse potential is given by the equation:

$V(x)=D(e^{-2\alpha x} - 2 e^{-\alpha x})$

It provides a relatively good approximation to a chemical bond. The graph of V(x) is shown below (for D = α = 1):

(Morse Potential Graph)

How would you expect the energy levels to be spaced for small n? What about for large n? Don't bother solving the Schrödinger equation (even though it can be solved exactly for this potential); just use your intuition. [Hint: For small n, consider energies less than around -0.8. For large n, consider energies larger than -0.5.]