Non-interacting electrons

Question

At the undergraduate level, in quantum mechanics, statistical mechanics, solid-state physics we always deal with non-interacting electrons.

I don't understand why we take non-interacting, and how it simplifies our lives. What are the consequences, if we not take non-interacting?

Also, what one understands from interaction here (in the context of quantum statistical) what sort of new activities happen?

If there are two non-interacting electron, it's like they are in two different universes. Then how does the Pauli exclusion principle act on them? I mean to say, if they are bound by some principle, then there should be some sort of interaction between them. I know the last question is quite silly.

Answer 1

Non-interacting electrons/particles is a useful model in order to explain the basic concepts in classical mechanics and quantum mechanics. It makes things much easier mathematically, since interacting problems can be rarely solved exactly (and cannot be solved exactly at all for more than three particles). Obviously, solving real-world problems requires taking interactions into account, but it also requires having advanced understanding of physics and math.

Statistical physics does not ignore the interactions! Rather it neglects them where they are not necessary and accounts for them phenomenologically when they are needed. E.g., atoms in a gas do not interact except when they collide. Thus, interactions can be neglected most of the time, as long as we take into account that it is these interactions that lead to statistical equilibrium! Similarly, it is the interactions that lead to equilibration between the system and the bath. These points are discussed in any good book on statistical mechanics, but are often overlooked, since these paragraphs/pages contain few equations.

Likewise, the solid state physics does not ignore the interactions either. Much of the basic solid state physics is however done within the effective mass approximation, which makes it look, as if the electrons are not interacting with the lattice, but which in reality is an elaborate approximation. Similarly, neglecting electron-electron interactions in a metal, which enables such simple derivations as, e.g., Drude formula, is grounded in the Fermi liquid theory - another rather advanced topic.

Answer 2

At the undergraduate level, In Quantum mechanics, statistical mechanics, solid-state physics we always deal with noninteracting electrons.

As a particle physicist I am not familiar with the term, and we do use a lot of quantum mechanics in particle physics. I looked it up:

We next turn to a discussion of noninteracting electrons, which we’ll define here as electrons that do not interact among themselves. Of course this is strictly speaking a fictional scenario. Nevertheless, there are several reasons why it is still worth discussing systems in which interactions between the electrons are neglected.

Italics mine

The text goes on to say:

Nevertheless, there are several reasons why it is still worth discussing systems in which interactions between the electrons are neglected.

So if you are interested you should read on, it has to do with modeling quantum mechanically matter in bulk.

You ask:

If there are two noninteracting electron , its like they are in two different universes.

If there are two electrons, there exists the Coulomb interaction between them . The term non-interacting is a term that says that the Coulomb interaction can be ignored to first order.

Then how does the pauli exclusion principle act on them?

The Pauli exclusion principle says they cannot occupy the same energy level. In systems where there are a lot of bound electrons, as in the conduction band of a solid for example, each one is at its own energy level, even though the difference is infinitesimal. The Coulomb interaction is taken care in the creation by nature of the energy levels.

This is the effect of quantum mechanics. Once the energy levels are defined in a collective potential, the electrons occupy successive layers according to the Pauli exclusion, which is a law, not an interaction.

In this sense the two electrons of the helium atom are non interacting, any effect of the coulomb potential between them has been incorporated in the energy levels which they occupy .