I Where do the vibrational modes of molecules come from?

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Where vibrational modes of molecules come from?
Hello everyone. First, sorry for my english. Second, I have got question where vibration mode of H2+ molecule (I think it is the most simple molecule for this topic explanation) comes from. If I should get basics before asking this tell me :). By my count the most important factor behind "being" oscillator is force that returns nuclei to the equilibrium position (I say about nuclei because I read in internet that in vibration are involved nuclei). As the motor force of nuclei repulsion (when nuclei are closer together than in eqilibrium position) I see Coulomb force, but if nuclei are further apart than in equilibrium position what force cause them to change direction and come closer to each other? In conclusion, what force (attracting nuclei) is counterpart of spring in classical oscillator?
 
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In a molecule, there is a balance between repulsive (electron-electron and nucleus-nucleus) forces and attractive (electron-nucleus) forces. The total force is zero at the equilibrium bond length. If you stretch or compress the bond, the imbalance of attractive and repulsive forces will pull the atoms back toward equilibrium.

This is probably as close an explanation as I can give you without invoking quantum mechanics.
 
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At large distances, you have attraction between a positive proton and a polarisable H atom. At small distances, the repulsion between the two nuclei dominates. Hence there must be at least one bound state and the distance of the two atoms can oscillate around the equilibrium distance.
 
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Not an expert in QM. AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order. But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical...
I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...

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