Pauli's Exclusion Principle - arbitrary?

[albroun]

Pauli's Exclusion Principle - arbitrary??

I have a puzzle about Pauli's Exclusion Principle, which, as far as I understand it, (being neither mathematical nor a physicist), states that fermions (as opposed to bosons) can never occupy the same space at the same time - well something along those lines - I know it is a bit more complex than that! But I have never come across any explanation yet as to why this should be so, just that it must be so because if it wasnt so, atomic structure as we know it could not exist. Is there a deep underlying reason (e.g. due to the spin values of fermions and bosons, or something else), or it just an arbitrary principle evoked more as a description rather than as an explanation?

Also if everything is indeterminate until the wavefunction collapses, can we then be certain that fermions conform to the Exclusion Principle, or is it only after the wavefunction collapses that we can really speak of fermions at all?

Sorry if these questions sound fundamentally confused - they probably are - just trying to get my head round this quantum weirdness stuff!

 

[The_Duck]

There is a deep underlying reason for the exclusion principle. It can be derived from first principles in quantum field theory (relativistic quantum mechanics); one finds that the theory doesn't make sense unless bosons have symmetric wave functions and fermions antisymmetric.

 

[albroun]

I am interested - can you point me to an explanation (written for a layperson) of the derivation in question? Or is it so fundamentally technical / mathematical that it is inexplicable in layperson terms?

 

[Matterwave]

There is a deep underlying reason for the exclusion principle. It can be derived from first principles in quantum field theory (relativistic quantum mechanics); one finds that the theory doesn't make sense unless bosons have symmetric wave functions and fermions antisymmetric.

For a layperson I wouldn't start here. I think a satisfactory start would just be that you name fermions the particles which have anti-symmetric wave functions, and the bosons the particles which have symmetric wave functions.

Basically, if you have 2 fermions together, their joint wave function must be anti-symmetric (by definition of fermions). This means that if you exchange the 2 particles (1->2, and 2->1) then the TOTAL (composite) wave function must change sign. If they occupied the same STATE (Pauli exclusion principle applies to states, not only to position - 2 fermions can be at the same position, but with different momentums and that's fine), then the wave function would vanish due to its anti-symmetry. Basically, if the two particles occupying the same state, you have that 1->2, 2->1 CAN NOT change the wave function, but by the anti-symmetric nature of fermions, this change MUST put a negative sign in front of the wave function. The ONLY way that BOTH of these (contradictory) conditions can hold is that if the wavefunction is simply 0 everywhere.

Where QFT comes in is to tie this definition of "fermion" vs "boson" to spin statistics. You will find in QFT that particles with half-integer spin are fermions, and particles with integer spin are bosons.

Without appealing to QFT, I think it's satisfactory merely to define fermions and bosons by the symmetric or anti-symmetric nature of their wave functions.

 

[Drakkith]

So the wavefunction, being anti-symmetric for fermions, means that if they occupied all of the same states they would cancel each other out?

 

[Matterwave]

I wouldn't really say "cancel each other out", it's more like, you can't, literally, construct an anti-symmetric wave function in which the particles occupy the same state. You have 2 contradictory requirements; you cannot fulfill both.

 

[albroun]

Thanks, but I think I am going to need a book or video that takes me through this stuff one simple step at a time. I don't know how symmetry applies to wavefunctions; and I have only a rather vague laypersonish idea of what a wavefunction is or what symmetry is in the first place, both concepts having very technical defintions in quantum physics and mathematics. The concept of spin is pretty esoteric too. I gather it is not to do with rotation in space, but something to do with the return of a particle to its original configuration. All will need a lot more explanation before I can begin to make sense of this stuff.

Any recommendations appreciated!

 

[Matterwave]

Griffiths chapter 5.1.1. gives a good introduction to this, but it assumes some prior knowledge of quantum mechanics (not much, but at least some of the formalisms), which are given in the previous 4 chapters.

 

[albroun]

Thanks - what is the name of the book by Griffiths?

 

[Matterwave]

Introduction to Quantum Mechanics