What is the basis for Pauli's exclusion principle?

[davelee]

Pauli's exclusion principle states that no two fermions may occupy the same quantum state. But why is this so? Is there a "why" explanation, as opposed to merely saying that this is the way it is?

 

[futz]

It has to do with the fact that for fermions, the wavefunction for two particles must be antisymmetric. If they were both in the same state, then the wavefunction would cancel out to zero, which does not make any sense physically.

 

[suyver]

Originally posted by futz
It has to do with the fact that for fermions, the wavefunction for two particles must be antisymmetric.

However, I think that this is an assumption and not something that one can prove from quantum mechanics.

For an n particle system one can construct (if memory serves) n! wavefunctions if the particles are indistinguisable. Now one assumes that only the totally symmetric and the totally antisymmetric wavefunctions will exist in nature and call the particles bosons or fermions respectively of which wavefunction they obey. So, on a fundamental level, I'd say that this is the case because it seems to work...

Anybody have any other thoughts?

 

[futz]

Originally posted by suyver
However, I think that this is an assumption and not something that one can prove from quantum mechanics.

At undergraduate levels, it is usually treated as such. However, the PEP (and for that matter, spin itself) is a direct consequence of relativistic QM, and falls naturally out of the theory.

 

[suyver]

Originally posted by futz
However, the PEP (and for that matter, spin itself) is a direct consequence of relativistic QM, and falls naturally out of the theory. [/B]

Cool, I didn't know that. So in the relativistic theory the fact that the wavefunctions must be either totally symmetric or totally antisymmetric follows from the theory? What causes all the other permutations to vanish?

 

[selfAdjoint]

Pauli originally gave the wave function explanation. Exclusion is now proved from the Wightman and other axioms for quantum theory. There is a famous but difficult book on this, "PCT, Spin, Statistics, and All That", by Streater and Wightman.

 

[lethe]

causal half integer spin fields must anticommute at spacelike separation, and causal integer spin fields must commute at spacelike separation, for the same reason.

the fields must be this way if they are to obey causality. if streater and wightman is too advanced for you (it is for me), then you can find a lower level "proof" of this statement in any quantum field theory book. i like weinbergs derivation a great deal.

and from this relationship between spin and commutation/anticommutation follows the pauli exclusion principle, for half integer spin particles.

so the Pauli exclusion principle is really a consequence of the union of quantum theory with special relativity.

 

[suyver]

Cool, I'll dig up Ryder's book 'Quantum Field Theory' over the weekend and have a look at that. Thanks for explaining! I feel rather bad that I forgot this, especially after learning quantum field theory from a Nobel-prize winner ('t Hooft).

 

[slyboy]

As correctly pointed out, the usual justification for the 'spin-statistics' connection is from relativistic quantum field theory. However, recently a number of authors have considered whether it is possible to prove the connection in ordinary non-relativistic quantum mechanics. There are a few different 'proofs' along these lines, most making heavy use of groups and symmetry properties in physics, and they remain highly controversial.

If you are interested, then most of the papers can be found on the quantum physics arXiv (http://www.arXiv.org/quant-ph ) by doing a search for 'spin statistics'.

 

[jeff]

Originally posted by jefferywinkler
Consequently, the wavefunction must vanish if the two fermions are located in the same state. Only 0 or 1 fermion can be there in one state, and this fact is essential for chemistry to work, for example, because we need the electrons to fill the different shells.

Hmm. Well, the exclusion principle explains the existence of the observed atomic orbital structure, and hence the very stability of matter.

 

[quantumdude]

Originally posted by jeff
Well, the exclusion principle explains the existence of atomic orbital structure,

You can solve the Schrödinger equation and determine the atomic orbital structure without ever encountering the exclusion principle. The PEP does not determine the orbitals, it determines how the orbitals are populated.

 

[jeff]

Originally posted by Tom
You can solve the Schrödinger equation and determine the atomic orbital structure without ever encountering the exclusion principle. The PEP does not determine the orbitals, it determines how the orbitals are populated.

I meant that without the PEP, there would be no orbital structure at all - whatever it's properties - since the orbit of electrons about atomic nuclei would decay. Thus rather than "existence of the observed..." I should've said "the observed existence...".

 

[lethe]

Originally posted by jeff
I meant that without the PEP, there would be no orbital structure at all - whatever it's properties - since the orbit of electrons about atomic nuclei would decay. Thus rather than "existence of the observed..." I should've said "the observed existence...".

why are you saying this? certainly it is true that without the exclusion principle, the ground state of an atom would have all electrons in the lowest orbital. but that doesn t mean that excited states can t exist, or that somehow the orbital structure is negligible.

the orbitals would all still exist, since the exclusion principle does not affect the solutions to the Schrödinger equation, they just wouldn t be populated for an atomic in its ground state.

 

[jeff]

Originally posted by lethe
...the exclusion principle does not affect the solutions to the Schrödinger equation...

I really didn't mean to imply anything about the role of the Schrödinger equation in atomic physics, either in relation to the PEP, or anything else. I apologize for my uncareful choice of words, but I was simply responding to jeffrey winkler's remarks about the relation between electron shells and the PEP and that without the latter "chemistry wouldn't work". I was making the point that this is an understatement to say the least, for without the PEP, atomic matter would be unstable! The chemical elements owe their very existence to the PEP! Thus the fact that we're here to argue about this right now would be impossible without the PEP!

Originally posted by lethe
...without the exclusion principle, the ground state of an atom would have all electrons in the lowest orbital.

Without the exclusion principle, orbits of negatively charged electrons about positively charged atomic nuclei would rapidly decay with the electrons plunging directly into nuclei and ending up sharing the same state. The PEP forbids such degeneracies.

 

[lethe]

Originally posted by jeff
Without the exclusion principle, orbits of negatively charged electrons about positively charged atomic nuclei would rapidly decay with the electrons plunging directly into nuclei and ending up sharing the same state. The PEP forbids such degeneracies.

i don t believe this. are you sure? if i found a selectron, let's suppose its stable and has the same mass and charge as the electron, only its a boson, could i not put it in the 1s orbital around a proton? what would cause it to spiral in?

doesn t this system still follow Schrödinger's equation? doesn't Schrödinger's equation have a lowest energy level solution for the hydrogen atom? the particle cannot go to a lower en energy level, completely independently of whether the particle is a boson or a fermion, it seems to me.

am i missing something here?

or, look at it another way, the bosons obey the Heisenberg Uncertainty Principle, right? so they plunge into the nucleus, but they can t stay there, and they won't be anihilated, so they come out again.

in fact, this is the same layman's argument for how Quantum Mechanics restored the stability of the unstable classical atom. i don t see how the fermionic or bosonic nature of the electron has any bearing on this argument.

 

[jeff]

Originally posted by lethe
i don t believe this. are you sure? if i found a selectron, let's suppose its stable and has the same mass and charge as the electron, only its a boson, could i not put it in the 1s orbital around a proton? what would cause it to spiral in?

doesn t this system still follow Schrödinger's equation? doesn't Schrödinger's equation have a lowest energy level solution for the hydrogen atom? the particle cannot go to a lower en energy level, completely independently of whether the particle is a boson or a fermion, it seems to me.

am i missing something here?

or, look at it another way, the bosons obey the Heisenberg Uncertainty Principle, right? so they plunge into the nucleus, but they can t stay there, and they won't be anihilated, so they come out again.

in fact, this is the same layman's argument for how Quantum Mechanics restored the stability of the unstable classical atom. i don t see how the fermionic or bosonic nature of the electron has any bearing on this argument.

Your right, and I know the origin of my temporary insanity. Anyway, what is true - not that it matters now - is that the PEP is responsible for the negative electron degeneracy pressure preventing atoms from approaching one another arbitrarily closely, thus ensuring the large-scale stability of matter. I'm quite embarrassed and am really sorry about this. Oh well, happy new year.

 

[lethe]

Originally posted by jeff
Oh well, happy new year.

no sweat. happy new years to you too.