How can the mechanism of electrons in an atom be explained? I am a high school student who takes both Physics and Chemistry.
Recently I learnt about the quantum mechanical point of view of looking at electrons or nuclei. I also learnt that the wave functions can be obtained by solving the Schrodinger's equation with various conditions specific to the problem (such as the particle in a box).
My shallow understanding of quantum mechanics is that we can only know the probability of an electron existing at a certain position and time, and the actual position can be determined when the 'observation' takes place.
The chemical bondings and chemical reactions are the results of electric interactions between nuclei and electrons. The Coulomb force is a function of the distance between two charges, so it is important that the exact locations of electrons should be known. But taking into consideration quantum mechanics, we don't even know where the electrons are, and we built up a subject called Chemistry, and most importantly, CHEMISTRY STILL WORKS VERY WELL.
So, what is going on?
 A: A basic difference between quantum mechanics and classical mechanics is that the potentials do not act on masses in quantum mechanics. Instead they are part of the differential equation that has to be solved to give the wavefunction for the system under consideration.
In the case of a single atom, lets take the hydrogen atom, the differential equation is simple, the potential enters the equation and the solutions, called 
 wavefunctions, come out; they show that the allowed locations are up to a lower energy level. The effect of the Coulomb potential in this case is to give the specific functions $Ψ$, which,   when complex conjugated   ($Ψ^*Ψ$) will give the probability of finding the electron at a specific (x,y,z,t), called an orbital.
For large atoms and aggregates of atoms again the same logic holds, that it is the potentials that have to enter in the quantum mechanical equation which will define the orbitals , i.e. where the electrons may be found . The classical attractive force becomes the quantum mechanical solutions of the orbitals. Effective models are sought in  order to describe the quantum mechanical behavior of many atoms in an ensemble, because of the many particle complexity. For example the band theory of solids.
A: It's true that because of QM you can't think of the electrons in the atoms as having precise positions. It's not just position that is affected by QM but all "observable" phenomena. You are right that a theory that explains forces as a function of position is therefore likely to run into problems. But there are quantum theories of how forces like electromagnetism work, even ones that take into account special relativity, and these have been used very successfully to describe what's going on with atoms. Your understanding of QM is not wrong, it's just that it's only the start of the journey.
A: In addition to the comments mentioned, when you solve for 2 electron problems in quantum mechanics, you do include a term of the form $\frac{kq_1q_2}{r_{12}}$ which represent the interaction between the two wavepackets. For more information on how the 2 electron system is solved, see https://en.wikipedia.org/wiki/Two-electron_atom#Schr%C3%B6dinger_equation. Moreover, without quantum mechanics, you can't explain phenomena like superconductivity and superfluidity. Even the transistors in your computer require quantum mechanics to work.
A: curios,
It is important to make a clear distinction between what is known or knowable and what it exists.
Quantum mechanics does not say that the electrons do not have precise positions or precise momenta. It tells you that:


*

*You cannot prepare a state where both the position and momentum of an electron are known with arbitrary accuracy (Heisenberg uncertainty principle).

*If the electron is not in a position eigenstate you can only predict the result of a position measurement probabilistically.
In other words quantum mechanics limits what can be known. It says nothing about what can exist.
Your observation that the classical Hamiltonian, based on Coulomb's law, works very well seems indeed to contradict the idea that particles do not have precise positions. It's not a rock-solid argument, but seems a little bit mysterious.
A: Chemistry and classical mechanics work well because they deal with statistical behavior involving many atoms and molecules, rather than individual particles. Due to the Law of Large Numbers, the overall behavior corresponds very closely to the probabilities calculated using quantum mechanics. So in many cases quantum behavior can be ignored, and classical models (like the planetary diagram of electrons orbiting around the nucleus) can be used -- it's an approximation (like all models), but it's normally close enough.
You generally only have to worry about quantum mechanical effects when you're dealing with very small numbers of particles. For instance, engineers designing microelectronic circuits have to deal with this, because they create "wires" that are just a few molecules thick. 
