# How can we demonstrate two particles can not occupy a same position at a same time?

## Main Question or Discussion Point

The Pauli principle say that many body fermi particle wave function is antisymmetry if we permute two particles.Then if the spatial wave function is antisymmetry,so there is not exist two particle occupy a same spatial coordinate at the same time.But I do not understand how about the circumtance if the spatial wave function being not antisymmetry(while the whole function still antisymmetry to satisfy the Pauli principle).I wonder this question because matter always occupy spatial volume.

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The Pauli principle say that many body fermi particle wave function is antisymmetry if we permute two particles.
There is actually loophole recently where you can have two fermi particles occupy the same space, but only in a special condition... doesn't quite answer your question, but it relates to the black hole horizon. I will try and find the paper for you. I found it quite interesting myself.

cgk

The Pauli principle say that many body fermi particle wave function is antisymmetry if we permute two particles.Then if the spatial wave function is antisymmetry,so there is not exist two particle occupy a same spatial coordinate at the same time.But I do not understand how about the circumtance if the spatial wave function being not antisymmetry(while the whole function still antisymmetry to satisfy the Pauli principle).I wonder this question because matter always occupy spatial volume.
As you say, the total wave function is anti-symmetric (note that in general it cannot be factored into a spatial and a spin part). That means that you have a zero amplitude for two Fermions having the same spatial AND spin coordinates at the *same time*. If either of those differs, there is no need for the wave function to vanish. In particular, this implies that two Fermions can occupy the same position in space--as long as they have different spin coordinates.

In fact, this actually happens even in the case of electrons with their singular Coulomb repulsion. For example, in molecules and solids opposite spin electrons have a non-zero and actually quite high probability of sitting right on top of each other.

Thank you very much for your answers!I still do not understand why electrons do not ''mix'' inside the nuclei of the atom,despite the nucleons and electron are not identical particles.

I have heard there exist neutron stars,then it is maybe electrons could fall into nuclei?

Thank you very much for your answers!I still do not understand why electrons do not ''mix'' inside the nuclei of the atom,despite the nucleons and electron are not identical particles.
Mix? Electrons can pass through the nucleus. It is also possible for an electron to be absorbed by a proton, though this doesn't happen very often.

Drakkith
Staff Emeritus

Thank you very much for your answers!I still do not understand why electrons do not ''mix'' inside the nuclei of the atom,despite the nucleons and electron are not identical particles.
The do mix. There is a finite probability of finding an electron inside the nucleus.

I have heard there exist neutron stars,then it is maybe electrons could fall into nuclei?
"Fall into nuclei" doesn't really mean anything. As I said, an electron can be found in the nucleus on occasion. It just doesn't stay there. During the collapse of a massive star in the supernova process, it becomes energetically favorable for electrons to "combine" with protons and form neutrons. That is pretty much what I would consider electrons "falling" into the nucleus if I understand your meaning correctly.

Then matter occupy spatial volume because Pauli principle and/or electromagnetic interaction and maybe partially(very small contributing) by weak interaction(?).Is that correct?

Quantum Mechanics ''spread'' many bodies in spatial volume by many spatial ''quantum orbits''?But how about the boson particles?

Drakkith
Staff Emeritus

Then matter occupy spatial volume because Pauli principle and/or electromagnetic interaction and maybe partially(very small contributing) by weak interaction(?).Is that correct?
Just the exclusion principle and electromagnetic repulsion. The weak interaction doesn't manifest as a "force".

Quantum Mechanics ''spread'' many bodies in spatial volume by many spatial ''quantum orbits''?But how about the boson particles?
Bosons don't have orbitals as far as I know. But this is only because they aren't electrically charged and don't have an atom to have orbitals in. They are still described by a wave function however.

Why we can not ''build'' the potential notion in weak interaction?Is it because the relativistic characteristic of weak interaction then we can not use the Born approximation?If so why we do not try to use notion ''relativistic classical force''?Because all interaction are the result of the exchange boson particles resulting in the change of momentum of interacting particles.

In general speaking there is not existing the notion potential in Quantum Field Theory because of the creation and annihilation of particles in QFT,therefore there is not the notion ''force''.Is that correct?