Adiabatic theorem and Born Oppenheimer Approximation

I try to understand the BO approximation here.

If we walk in mountain, some times, quit a bunch of mosquitoes like bees would fly around your head. You keep on hiking but the bees follow you by swirling around your head (and making noise too...really annoying).

Now, in molecule or materials, the nucleus can be given analogy of your head. The electrons can be given an analogy of bees around the head of a hiker. Here, the movement of the hiker's head (nucleus) does not affect the movement of bees (electron movement). This analogy may be useful for a kindergartner. But, for me, it is not enough. How to understand this really? Let us break this big problem in to pieces.

Another usual textbook discussion is the following. The mass of proton is 1837 times heavier than electron. Thus, nucleus is much much heavier than electron. Because of this, we can consider the nucleus is fixed in space and electrons are the only moving particles. This also doesn't makes sense to me. Just the higher mass consideration is not a satisfying argument to me.

So, I need a more intuitive as well as rigorous argument.

Let us start with Adiabatic theorem.

If a quantum system is changed gradually, the final state has the same state as the original state. Griffiths Quantum Mechanics beautifully explains this quantum adiabatic theorem. What if we change the system drastically?

Now, come to Born-Oppenheimer approximation. In BOA, we make use of this adiabatic theorem. Right? Where it is used? How it is used to prove it in original paper? Is there a simpler version (text book version) to understand the proof now?

Relation between statistical adiabatic process and quantum adiabatic theorem

The adiabatic process which we learn in thermodynamics is the process where we decrease/increase the volume of the gases enclosed in a container, slowly so that the pressure remains constant throughout the process.

The quantum mechanical adiabatic theorem says: For a system with non-degenerate ground state, if the system is taken from the ground state to the next (only next excited state) energy level (i.e. excited state) very slowly (the time taken to go from one state to another in a smaller time than the characteristic time of the energy gap of the two energy-levels involved), then the system finally adapts ground state.

Here analogy is as follows (even though it doesn't make sense now)

• Which remains constant: Pressure & Energy
• Which is changed: Volume & State of the system
• In what time : Both are slowly (in QM, it is exactly defined; In TD, not defined (?) clearly.

Adiabatic theorem plays an important role in the definition of topological phases of matter.

In approximate (but most successful) theory such as Density Functional theory, construction of the final state is obtained by slowly changing some  parameters (for e.g., λ) and finally the ground state is obtained. 

What are the connection among these? 

Another similar idea (?) is the derivation of molecular forces by Feynman. What is the exact connection among these?