Sketching a Phase Portrait for Diatomic Molecule

[Amok]

Homework Statement

I have to sketch a phase portrait (in phase space) for a system which is a diatomic molecule. The effective potential is equal to a potential that is attractive at long range and repulsive at short range plus the centrifugal barrier. It basically looks like a Morse potential with a little "bump" right outside the potential well (due to centrifugal barrier).

Homework Equations

$$V_{eff}(r)=V(r)+ \frac{L^2}{2 \mu r^2}$$

mu is the reduced mass, r is the distance between atoms and L is angular momentum (a constant)

The Attempt at a Solution

I wrote down Hamilton's equations (assuming a Morse or Lennard-Jones potential) and tried to put them on an applet for this kind of stuff but the results seem weird. For low energies I think the system will have an oscillatory motion (so ellipses will appear in phase space), but I'm kinda clueless about higher energies. I guess that given enough initial potential energy r will simply increase constantly and p will go through one maximum and one minimum. The equilibrium point will be the distance corresponding to the bottom of the well and the top of the bump. Can I get some help?

 

[mark726]

Hi there,

Thank you for your post. It sounds like you are on the right track with your approach. In general, phase portraits for diatomic molecules can be quite complex, especially when considering the effects of the centrifugal barrier. One approach you could take is to first plot the effective potential as a function of distance between the atoms (r), and then overlay the Hamiltonian contours on top of this potential.

For low energies, as you mentioned, the system will exhibit oscillatory motion and the phase portrait will look like a series of ellipses. As the energy increases, the system will exhibit more complex behavior, including the possibility of chaotic motion. In general, at higher energies, the phase portrait will be more spread out and less regular.

Another approach you could take is to plot the phase portrait for a range of energies, starting with low energies and gradually increasing them. This will give you a better sense of how the phase portrait changes as the energy increases.

One thing to keep in mind is that the phase portrait will also depend on the specific form of the potential you are using. A Morse potential with a bump outside the potential well will have a different phase portrait compared to a Lennard-Jones potential with the same bump.

I hope this helps! Good luck with your phase portrait.