Why do batteries in series add up in voltage? (real explanation requested) This has been asked before but I need to be clear on what kind of answer I'm looking for. All the other posts are just water analogies.
I get it, you add the voltages together. And there's a magical unicorn pumping it aka the water pump and hill-tilts on hill-tilts with meatballs rolling on meatballs.
Seriously though, how does it REALLY work? I don't need any analogies to tell me WHAT it does, because I can sum that up in a simple formula. And I really really do not want any false quasi-explanations.
If you must use analogies to convince yourself of its workings, you simply do not understand it. Electricity is NOT water.
So if we have two batteries of the same chemistry in series, why do we get higher Fermi levels? The chemistry shouldn't support that, the chemistry has a set Fermi level: you get 1.5V of some materials. But somehow, by having them in series you get e.g. 6V. How does that work in terms of particle physics? That is very crucial here: how do we chemically and electrically get that higher fermi level.
 A: ... full answer can be found in Fundamentals of Electrochemistry
edited by Vladimir S. Bagotsky
short answer:
all electrochemical batteries have the same working mechanism: either anion is reduced at the cathode terminal and, acting as the charge carrier, diffuses into electrolyte and then migrates to anode to engage in oxidation reaction
OR 
cation is oxidized at anode (releases high energy electron into anode material) and then migrates across electrolyte via diffusion to cathode and is reduced
... either way, the energy of the electron(s) involved in the redux reactions is conserved: connecting batteries in series simply means that the energy level of electrons available for redux reactions is higher to start with at the anode (and lower at the cathode)
and once you close your a circuit (make a conductive connection between cathode and anode terminals) you'll experience what is referred to as electric current: electrons will flow from high potential to lower potential, releasing energy (voltage) in the process
A: Voltage is a potential difference.  The difference here is important.
Firstly ill explain potential energy in general.  Lets look at gravitational potential (which is behaves in the exact same way).  It is given by $mgh$.  But what is $h$, is it measured from the centre of the Earth, the surface, the floor, the shelf?  The truth is, it doesn't matter.  When you use it, all that matters is the difference in potential energy, not the actual value.  This means that energy is only defined up to a constant:  $mgh$ will work just as well as $mgh+392.5$, the constant doesn't matter.
I know you said not to use analogies but gravity and electro-statics are very similar (look at coulombs law and newtons law).
A potential is just defined as the potential energy per unit charge (for electrostatics) or mass (for gravity).  So almost anything I say about potential energy will be true for potentials in general.
So the battery doesn't create 1.5v, there is no such this as a set potential of 1.5 volts.  What it does create is a potential difference of 1.5 volts.  
Lets say I have a pole of 1.5m.  The potential difference between the ends, if held vertically, is $1.5gh$.  If I move that up or down the potential difference doesn't change.  I can stack several on top of each other and we can, quite obviously, just add the lengths.
batteries are no different - mathematically a battery is just the same as holding a ball slightly higher up.
In short, potentials/energy is only ever a difference, there is no universal 1.5v.
Really good question by the way.  If you want to know more, look at an a-level physics book on gravity and electro-statics.  It should also have a section on the similarities and differences and you will see then how energy really works.  
A: Here is my hypothesis. I was imagining a charged lithium ion cell in series with another (please watch those video to understand my lithium ion batter references: https://youtu.be/VxMM4g2Sk8U) . When the circuit is complete, I imagine electrons move from the graphite layer into the metal foil carrier up to the cathode from there they go thru a wire to the other cell’s anode and (scenario 1) either stay in the lithium oxide complex somewhere just sitting there kinda or (scenario 2) move over through the second cell’s electrolyte layer to the it’s graphite layer where it stays along with the electrons that were already there prior so now more electrons ready to jump through cathode of second cell. While in the first cell because it’s electrons already left, the lithium ions are free from the graphite complex and would move towards the electrolyte layer to the lithium oxide complex layer under the anode of first cell. Now there would be a large positive charge here due to the positive lithium ion build up in first cells anode and larger negative charge second cells cathode so if a wire was connected between those 2 to complete that circuit as I mentioned earlier then there would be a greater potential aka a greater voltage. Now all that was the second scenario. Let’s talk about the first one from earlier I put in parenthesis as well. So after the electrons just kind of chill in the lithium oxide structure in cell 2. the electrons on the other side of electrolyte layer in cell 2 will move through cathode through wire over to the first cell’s anode where the electrons settle in the first cell’s lithium oxide structure. Since cell 1 lost its electrons as mentioned in the beginning it now has electrons in the oxide structure and positive ions in the graphite structure and those electrons which were lost by cell 1 that are now in cell 2 are in cell 2’s oxide structure. So now I’m an overvie overview of scenario 1, we have electrons in the oxide structures of both cells and ions in the graphite structures of both cells. From there the ions will go through the electrolyte layers in both cells and absorb the electrons sitting there and bond covalently within the oxide structure. So now there is a potential created by the “wanting” of this reaction to happen, getting the electrons to a stable place and the overall system to stability. With one cell it’s just the potential of trying to move one set of electrons to the oxide structure and bond up and all that but with 2 cells, the potential is now the combination of both cells to get both sets of electrons in each to make that move to stability and bond up. In this explanation I break things into step like when I say the electron first go here and sort of wait there but in truth they are always moving in fluid motion and not in that choppy manner.  I just  say it that way to simplify it into more easily comprehensible steps. So in the second scenario when I say the electrons in cell 2 just kind of ‘move to the graphite structure and pile up with the other electrons, I mean it more like the electrons initially there already started moving through the cathode into cell 2 through a power consuming utility (if there is one, if not then just back into the other batter ( it’s gonna heat up and short circuit then but meh)) and the electrons that just came from cell 2 are in pursuit of those cell 1 electrons that just left back to cell 2 because they are heading to a large positive charge. Also in scenario 1 I don’t identify a direct charge difference in the end that causes voltage like in scenario 2 (for this reason I favor 2) but I rather say it like the whole over system’s reaction including all it’s aspects and steps that cause the potential (voltage). To simplify It’s like the electron is thinking, “oh ok I see if I do that and go there and do all those thing I finally reach a position of rest” like the electron can see the future and know how to go, so a potential is created because the electrons can go from a high energy state to a lower one. I used voltage and potential interchangeably, sorry lol. I am just a high school kid so an amateur. I just want to know if this theory even slightly makes sense.
A: Here's my hypothesis. Assume batteries A->B. Orientation: + -
The drain from A to B is no mystery. A- and +B gets into equilibrium, both change by a half in potential. That does something chemically (which I'm curious about) which drops +A 1/2 in potential to have some kind of chemical balance with A- (that just dropped). Same happens for B, and you get an extra 2*(1/2)=1 (battery "unit") potential difference.
To clarify, the chemistry of the battery is such that there's a set (e.g. 1.5) potential difference as a minimum; anything less and it's imbalanced and reacts and creates a 1.5 difference.
I will have to verify this theory. 
