How can the mechanism of electrons in an atom be explained? [closed]

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?

• Probability applies to everything, including force (and velocity, and momentum, and angular momentum, and energy, etc.). It’s not just position that is probabilistic. – G. Smith Nov 22 '19 at 8:21
• Chemistry is as it is because not in spite of quantum mechanics. – my2cts Nov 22 '19 at 8:23
• In nowadays Chemistry is based on quantum mechanics as well, just google "molecular quantum mechanics". So, Chemistry is not contradicting to QM, it's in reverse - QM knowledge is applied to Chemistry as well – Agnius Vasiliauskas Nov 22 '19 at 8:23
• Often you can treat electrons in an atom as a cloud of charge whose charge density is proportional to the probability density for the electron. This cloud of negative charge bonds the positively-charged nuclei of molecules together in a quasi-classical way, since the nuclei are heavy and “less quantum-mechanical” in some sense. – G. Smith Nov 22 '19 at 8:26
• It's like how the half life of a single atom has a statistical distribution, but when you are considering grams of a substance you can ignore the distribution and focus on the average time it takes for something to decay -- there's just so many atoms in a gram of something that the nuances that happen on a single atom level are smoothed out. Chemistry can ignore all of the weird stuff that's going on under the surface because all that stuff is abstracted away. – eps Nov 22 '19 at 18:34

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.

• Yes. I just googled it and found that it is quantum electrodynamics, largely contributed by Richard Feynman (according to Wikipedia). I just gave up understanding this as I realised that it's not the kind of thing that I can understand. Thanks to your and the others' answers, I at least clarified that there somehow is a way to deal with such interactions in terms of quantum mechanics, which is a good thing. – curious Nov 22 '19 at 9:42
• Yes that is the right theory. You certainly can understand it though and I would encourage you to try. As you've just figured out QM has a lot of consequences and challenges our whole perception of reality so you can't really afford to just look the other way :) Start with Leonard Susskind's Theoretical Minimum book. – user68014 Nov 22 '19 at 9:49
• It is not true that the problem is in the fact that forces depend on position. The non relativistic Schrödinger equation provides an accurate description of the properties of all the light atoms even though the potential energy term is a function of the position. It's the passages from a position-dependent interaction to the observables which is peculiar and makes the difference with classical physics. – GiorgioP Nov 22 '19 at 10:10

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.

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.

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:

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

2. 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.

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.