[Schrödinger]

Please take a look at the image below.

Consider a ball and spring model of a molecule (say diatomic). I understand that when you do not account for the KE that arises out of Heisenberg's uncertainty principle (i.e, energy is fully potential) and we assume that the atoms are being brought closer to each other from infinity, we get the Lennard-Jones 12-6 potential graph.

But in reality, since Heisenberg's uncertainty applies, we have to take into account for the KE of the molecule as well and so, we have a certain Zero Point Energy.

And as the graph suggests, there are different vibrational levels that the molecule occupies, depending on the temperature and vibrational quantum number. I got stuck trying to interpret this one... I'd be grateful if someone could point me in the right direction.

My doubt is : what exactly happens as the atoms are brought closer? Look at the dotted line (L-J 12-6 potential graph). As the atoms are brought together, the energy passes through a minimum corresponding to r_e and then rises as expected. So what exactly happens at the places where these straight lines are drawn (the vibrational levels) ? Does the energy take the red path? Why are these straight lines corresponding to different vibrational states drawn? What do they signify?

[voidSetup]

I'm not sure if this is what you're looking for, but do the straight lines correspond to changes in the bond length as a result of the change in vibrational state which is higher in energy?

[MrTeo]

I think the confusion here comes from the fact that you're considering an intermolecular effect such as the LJ potential and the harmonic vibration of a bond as parts of the same physical effect. When two atoms are brought together you will see the the usual curve (caused by the LJ potential if the two atoms are neutral: obviously if we talk about ions the whole thing changes) which takes into account only the attractive and repulsive effects of the nuclei and the electron clouds, and then a minimum when you reach the average bond lenght of your diatomic, just to makes things easier to explain, molecule. Now, the harmonic quantistic oscillator (the parabolic curve is in fact an approximation of the LJ curve around the bottom, considering an harmonic behaviour... sometimes also a term related to anharmonicity must be added) tells us how the molecule vibrates around the equilibrium position (the bottom of the curve, so the average lenght), and the important thing is, first of all, that there is no such thing as a completeley still atomic system (as you correctly pointed out the zero energy is not 0), and also that there's not a continuous energy change from one state to the other, but well defined steps (the lines) with constant energy differences: the old refrain that the energy is quantized. If you give the molecule enough energy the amplitude of the oscillations will increase until you'll reach a point where you will break the whole system.

Hope it's all clear now...

[Schrödinger]

So does that mean that the atoms actually oscillate with an amplitude defined by that red horizontal line in the ground state? And when pushed closer together/farther away than the amplitudes allowed, the energy deviates from the straight line and takes the curved path?

[MrTeo]

So does that mean that the atoms actually oscillate with an amplitude defined by that red horizontal line in the ground state?

Yes and no: we're talking about quantum armonic oscillator so probably the lenght of the horizontal lines has some kind of meaning linked to the amplitude (e.g. when the molecule breaks the "amplitude" goes to infinity) but I don't exactly know what it is. Still need some QM to fully understand this sort of things...

And when pushed closer together/farther away than the amplitudes allowed, the energy deviates from the straight line and takes the curved path?

I think you continue linking two things (the different energetic leves of the quantum oscillator and the classical LJ potential) in the wrong way: if you push the atoms closer and then you cease the interaction which caused this changement I think that they'll briefly return to their equilibrium position (through a series of oscillations I'd say) and maybe the vibrations of your system will change or also overlap with the new ones, but I cant' really say how they interact. If you continue applying your "force" you'll have a new equilibrium position and a different curve, but the molecule will go on oscillating as usual (though I don't know if the frequency will change).

Seems to me that the confusion arises from being on the interface between classical and quantum: thinking about the molecule as a system of two balls oscillating around the equilbrium can be hepful, but what you "really" have down there is a wavefunction.

[Schrödinger]

Yes. I guess I have been linking the classic LJ potential and QM : the reason being the confinement of those straight lines to within the parabolic curve. I mean, if there is no relation between them, then why at all would the straight lines be 'line segments'? They are after all energy levels, and the shortened length leads me to believe that there is some sort of connection between the bond length and the straight lines.
It's like as if the bonds vibrate about the mean position in an amplitude (symmetrical in the case of the quadratic parabola and unsymmetrical in the case of LJ potential graph) defined by the length of those segments. Probably that's where I got confused.

[MrTeo]

I tried to fix up some of the gaps I left in my explanation, here's what I found out:

The LJ potential creates the boundaries of our potential energy hole. "Confined" (but not completely, and here some messy topics about QM and the possibility to reach regions with E<V pop out) by these boundaries are the energy levels (with all what I said about them) and there we have our wavefunctions through which, if we square them and take the abs value, we can know the probability distribution of the electrons. The levels are only representation of the energy spacing, to be precise we should graph something like this: http://en.wikipedia.org/wiki/File:HarmOsziFunktionen.png

Moreover looking at this graph we can easily see the correspondence between classical physical intuition and the QM formulation of concepts:

Note that the ground state probability density is concentrated at the origin. This means the particle spends most of its time at the bottom of the potential well, as we would expect for a state with little energy. As the energy increases, the probability density becomes concentrated at the classical "turning points", where the state's energy coincides with the potential energy. This is consistent with the classical harmonic oscillator, in which the particle spends most of its time (and is therefore most likely to be found) at the turning points, where it is the slowest. The correspondence principle is thus satisfied.

So as you can see the length of the lines representing the energy doesn't even tell us exactly (if we can talk about "exact" values at this scale) the amplitude of the oscillation, but we can find that out observing the areas (points) at which the particles are more probably found (just like with a mass and spring system): the turning points.