# Pauli exclution principles.

1. Jan 27, 2005

### somy

I had a question about the pauli principle. Well, I know that it classify the particles into two groups named bosons and fermions and says that they have symmetrical or antisymetrical wave functions.
My question is this:
How did he say these? I mean, was it assumption that came true or just a fact that was observed. And if so, what was the exact experiment and its resoning?
Somy

2. Jan 27, 2005

Staff Emeritus
Basically it works from the wave functions. The boson wave functions are symmetric in that tif you make a spatial reflection they don't change sign. Fermions have antisymmetric wave functions, they go into their negatives if you make a reflection. Now it isn't obvious but this means that if you combine two wave functions to represent two particles in the same state, the boson wave functions will add, giving you a new wave function for the combined state, and from that you can calculate the probability of that combined state. But the fermion wave functions subtract, giving zero, so the probability of a combined state is zero.

This is way oversimplified; to prove the theorem rigorously is actually pretty difficult. There's a famous book called Spin, Statistics, and All That by theory mavens Streator and Wightman that proves it; it's been the downfall of many a smart student who thought he could learn it cold.

Pauli himself got a real proof, not as complete as S&W's, but good enough to convince him, so he announced the exclusion principle.

3. Jan 27, 2005

### da_willem

Pauli's proof that relates the spin of a particles to it's statistics, the spin-statistics theorem is quite advanced and long. This made Feynman come to the thought that probably we don't have the necessary understanding yet of the principles involved. He was convinced that for such a simle rule should be a simple explanation. But some decades after he wrote that down there is still no easy way of seeing the relation between spin and the behaviour of particles like the Pauli exclusion principle.

4. Jan 27, 2005

### somy

Can you explain more dear da_willem???
I mean the feynman's work.

5. Jan 28, 2005

### da_willem

Feynman Lectures on Physics III p4-4 :

6. Jan 28, 2005

### BlackBaron

Actually, just a bit before he died, Feynman did find an "elementary proof" to the spin-statistic theorem (I use quatation marks because I've seen some arguments against that proof, but nevertheless, is good enough to me ), he explains it in the first Dirac Memorial Lectures in 1986 at Cambridge University (Feynman died in 1988 ).
In that lecture he also explains the reason of antiparticles and Dirac's magnetic monopole. There is a book with that lecture together with the second Memorial Lecture by S. Weinberg (where he talks about symmetries). I've also found some papers in the American Journal of Physics that talk about that proof, sadly, I don't have any of those references at hand, but I'll look for them.
The proof has 3 parts:
First, you show that when you rotate a half-spin particle (fermion) 360° its wavefunction changes sign (on the other hand, there's not sign change when you rotate integer spin particles (bosons)). This is explained in many quantum mechanics textbooks.
Second, you show that when you exchange 2 particles there's implied a 360° rotation that produces the minus sign and that's what causes the antisymmetry of the total wavefunction (for bosons, there's no sign change in the 360° rotation, and that's why their total wavefunction is symmetric). This is the crucial step of the demostrations and the one I've only seen in those few places I mentioned before).
Third, and finally, you relate the antisymetry of the total wavefunction with the exclussion principle the way selfAdjoint explained. This is also explained in many quantum mechanics books.

As for the usual proof of the spin-statistic theorem, it's briefly sketched in the wonderful book "The history of Spin" by Sin-Itiro Tomonaga (who, by the way, shared the Nobel prize with Feynman and Schwinger), along with the few things you need to know about quantum field theory.
Basically, you first show that if you quantize fermionic fields (wich are related to half-integer spin particles) with anticonmutators you get a consistent theory, while if you use conmutators, you don't; the exact opposite happens with bosonic fields (wich correspond to integer spin particles), you have to quantize them with conmutators instead of anticonmutators, otherwise you get an inconsistent theory; showing those two facts are the difficult part of the proof. Then, you see how the (anti)conmutators are related to the (anti)symmetry of the wavefunctions when you exchange 2 particles (this particular fact is very well explained in the book Baym, G., "Lectures on Quantum Mechanics", Addison-Wesley, 1969, wich, by the way, I like very, very much ).

I hope that helps.
Cya

7. Jan 29, 2005

### somy

ok!!! then it has a proof!!! I think I get my answer, but too early to understand the exact proof!!! (I have to study a lot!!!)

8. Feb 5, 2005

### mikeyork

There have been many such simple proofs. Unfortunately they almost all suffer from the same defect as the "official" field theory proofs. They rely on proving a statement of the theorem that is, at best, ambiguous and more bluntly, wrong. They are therefore incomplete.

That statement is the one that relates symmetry or anti-symmetry of the wave function to the particle spin. The correct statement (and the one which is observable and verified in nature) concerns the general identical particle exclusion rule (odd values of composite spin are excluded when all other quantum numbers are identical) -- which is, BTW, independent of the individual particle spin. The usual distinction between two different "types" of particle is then spurious.

In May 2000 I presented the first complete proof of the theorem at the Spin2000 conference in Italy. It is published in the proceedings ("Symmetrizing The Symmetrization Postulate", Michael York,AIP Conference Proceedings vol 545, pp104-110.) and is downloadable from the Xarchiv (http://xxx.lanl.gov/abs/quant-ph/0006101).

The proof assumes only basic quantum mechanics together with the (obvious) assumption that particle permutation is an artifice of the way we construct state vectors and is not observable. "Exchange" of identical particles is an example of such a permutation.

The elements of the proof are:

1. The exclusion rules are an interference effect. To compute such an effect one must use single-valued state vectors (i.e. eliminate arbitrary relative phases). This in turn requires physically complete state descriptions.

2. If one uses permutation symmetric (PS) state descriptions to uniquely identify single-valued state vectors, then the state vector must be permutation invariant (i.e. exchange symmetric) or involve an artificial and arbitrary order dependent phase that is physically inconsequential. This applies independently of the spin. (Although I make a detailed argument about this, it is essentially obvious, because a single-valued state vector that is unique to a unique state cannot change its phase just because one uses a different but physically equivalent description of the state.)

3. Given any two particles, one can choose spin quantization frames (SQFs) in a way that is symmetric or asymmetric between the two particles. In the symmetric case one can define PS state vectors. In the asymmetric case, permutation can introduce an asymmetry into the state description and therefore into the state vector. But this permutation (or "exchange") phase can always be obtained by relating to the PS case. It turns out that a particular case (that just happens to be the one unconsciously chosen in all proofs that depend on proving antisymmetry for fermions) is permutation antisymmetric for half-odd-integer spin particles because the relationship to the PS case involves a 2pi rotation on one particle's SQF relative to the other's. But the fact that one can do this is actually irrelevant.

4. The exclusion rules are then proved simply from the symmetry properties of Clebsch-Gordon coefficients (i.e. the rotation group) taking care to use unique specifications of the SQFs.

9. Feb 5, 2005

### dextercioby

Are u claiming that the Symmetrization Postulate can be logically deducted/built starting with another (in this context proving themselves more fundamental) principles/premises ??

If so,and you're right,then that's another postulate of the QM in the traditional/Dirac formulation which would prove itself not ABSOLUTE.Have you tried to extend your calculations over the von Neumann's formulation?

Daniel.

10. Feb 6, 2005

### mikeyork

Not the Symmetrization Postulate (which claims anti-symmetrization for fermions and is irrelevant) but the exclusion rules. And all you need to start with is basic QM.

I don't know what you mean by "ABSOLUTE". If you mean that it doesn't need a separate postulate, then I agree.
No. I've never gone into the difference in formulations and I doubt that they are relevant. The proof really doesn't need much more than the usual superposition principle because it is a straightforward interference effect.

Last edited: Feb 6, 2005
11. Feb 6, 2005

### dextercioby

You got me lost now.Your article's title refers to the Symmetrization Postulate... What do you mean by "exclusion rules"...(do they have anything to do with the "exclusion principle",you know that initial formulation of this principle due to W.Pauli (1925) and which applied to fermions only...??) ?What do you mean by "basic QM"??Isn't the Symm.Principle taught in schools assumeing "basic knowledge" of QM...?

No.Why would the Symm.Postulate need another postulate??By ABSOLUTE i meant just what any mathematician needs as an adjective to associace with the words "axiom"/"postulate" (i.e.which cannot be proved,ergo "absolute"...).

I wouldn't call the difference between the 2 formulations as "irrelevant".von Neumann's is much more elegant and honestly much more useful in quantum statistics.

Daniel.

12. Feb 6, 2005

### mikeyork

The title "Symmetrizing The Symmetrization Postulate" is a pun that indicates certain characteristics of my proof of the general exclusion rule and emphasizes how my proof differs from the usual formulations.
Yes. But there are exclusion rules for all identical particles, not just electrons. It just happens that they can be expressed as a single general rule that odd composiite spin is not allowed. Composite spin is the angular momentum obtained by adding the spin of the two particles. The Pauli principle is just a special case where for two electrons the composite spin must be 0, because 1 is odd and the composite spin for two spin half particles cannot be greater than 1. The more general rule manifests in particle scattering, for instance, when you collide two identical particles (or near-identical particles if you ignore symmetry-breaking).
The Symmetrization Postulate is the rule that bosons are symmetrized and fermions are anti-symmetrized. This is an extra postulate over and above basic QM. What I show, amongst other things, is that both fermions and bosons can be described more naturally in a symmetrized form and that this requires no extra postulate, because it is implicit in QM if you accept that particle permutation has no physical implications. And that is the point of my pun: the old postulate that everyone tried to prove is irrelevant, since it is either ambiguous (depending on exactly how you describe two particle states), or just plain wrong (if you use what I think is the more natural permutation-invariant way of describing two particle states)

I don't doubt you. What I intended is that I suspect it is irrelevant to my proof.