How to formulate a mathematically precise Pauli exclusion principle?

The standard way of stating the Pauli exclusion principle is that two fermions cannot "share the same quantum state". As we noted in the Nov 20, 2018 blog post, the concept "the quantum state of a single electron" is unclear if many electrons are present and interact.

The Pauli exclusion principle was formulated by Wolfgang Pauli in 1924 to explain the placement of electrons in orbitals in many-electron atoms. Empirically, the electrons seem to fill the pigeonholes that we find when we solve a one-electron atom, the hydrogen atom.

Rather than proving a general exclusion principle, whose statement is unclear, we can start from a much more concrete problem.

Problem 1. Prove that electrons in many-electron atoms, in some sense, fill the orbitals of the hydrogen atom in a way where just one electron can occupy one orbital of hydrogen. An orbital of hydrogen is determined by its four quantum numbers n, l, m, s.


Problem 1 contains the phrase "in some sense fill the orbitals", which is unclear mathematically. When someone solves Problem 1, he should also clarify what that phrase means.

Problem 1 is a many-body problem, and such problems are notoriously hard to solve. A brief Internet search does not reveal any picture of the orbitals of helium. Even the simplest case of two electrons remains unsolved.

Our own heuristic derivation of the Pauli exclusion principle in the Nov 24, 2018 post is by no means a mathematical proof. Furthermore, it does not explain why electrons should fall in the pigeonholes determined by the single electron in the hydrogen atom. We can present a more modest problem:

Problem 2. Give a heuristic argument which explains why the orbitals of hydrogen determine the pigeonholes where electrons fall in a many-electron atom.


Solving the many-body problem of mutually repulsive particles might be easier for particles in a box. A step forward in solving Problem 2 is to find any setup where a heuristic analysis shows that electrons fall into the pigeonholes determined by the ground state and the excited states of a single electron system.

Our Nov 24, 2018 post does give some clues that repulsive electrons in a box might fill states that resemble single-electron states.

The spin-statistics theorem

The spin-statistics theorem is claimed to imply the Pauli exclusion principle. The spin-statistics theorem states that the "sign of the wave function flips at an interchange of two fermions".

Since in quantum mechanics, one is allowed to multiply an arbitrary wave function by any exp(i φ), where φ is real, what prevents us from flipping the sign of the wave function back?

Also, the "interchange of two fermions" is not clear. What does that mean physically and how can we be sure the two fermions were interchanged when fermions are indistinguishable?

In most sources, "interchanging" seems to mean that we change the order of two creation operators in our mathematical description of the system. The creation operators are applied at spacetime points x and y where the separation of points is spacelike. If the creation operators would be real physical events, then there is no physical ordering of them in special relativity, and consequently, it makes no sense to "interchange" the order of these events.

But it may be sensible to interchange the order of operators in our description of the system. Can we calculate something by manipulating our description of the system in such a way?

Is there an empirical experiment where particles are "interchanged" in some sense, and the interference pattern shows that their wave function flipped the sign?

http://www.feynmanlectures.caltech.edu/III_04.html

Feynman's lectures contain a diagram of scattering experiment, where the interchange depends on how close the particles come to each other. It is not just an interchange, but the paths of the particles in their mutual potential differ in a significant way.