# Why Pauli's Exclusion Principle?

1. Jul 15, 2005

### TriTertButoxy

Hi. I'm a sophomore undergrad studying chemistry at Rice University. I recently took quantum mechanics... and I now realize I should have gone into physics rather than chemistry. Ok. Whatever.

Bosons are particles with integer spins (helium-4 atoms). Fermions are particles with half-odd integer spins (electrons). I can't think of a non-(God-made-it-so) reason for why every fermion in a given system must have a unique set of quantum numbers (Pauli's Exclusion Principle). This rule simply doesn't apply to bosons. While this phenomena in nature is the very reason we observe such diverse chemistry among the chemical elements, could it be that the exclusion principle is simply an axiom of nature?

If not, I'm really, really dying for a good expanation.

2. Jul 15, 2005

### Spin_Network

[/I]
In a Nutshell Proton Decay! ..in an ElectronShell Virtual-Decay

To get to the bottom of things you have to start from the top

If one starts from topDOWN principles, then the Electron Spin, prevents tables and chairs from penetrating the floor they stand upon.

The 'Space-Quantization', or by the same person who coined this effect, Pauli Exclusion Principle, is generated from Electron Spin Angular Momentum.

Quantum Ranging?..Whilst moving in a certain direction, say along the Mendelev Table?..one has to know which way is up, and which way is down?, with respect to Atomic Structure, things have to know along which 'Path', Gravity wants to send its signals!

Photons comminicate this fact to Neigbouring matter , without this communication, jumping off a Table to the floor would be rather boring immpossibility!

The Uniqueness of the Quantum Numbers is contained by Geometric Paths, Triangles are Different to Spheres for specific reasons, not axiomic, if a Table was a perfect Orb/sphere, it could not be balanced upon a Perfect Triangluar Floor, the specific shape induce's movement, over-distance at specific rates.

Last edited: Jul 15, 2005
3. Jul 15, 2005

### da_willem

If you keep asking why to a question, sooner or later you will come to a question that is (yet) unanswerable. As this is quite a fundamental question, you will come to this pretty fast.

Maybe you already know this but I'll explain where the Pauli principle comes from, in which I will assume you know what a wavefunction is.

If you have two distinguishable particles a and b (a proton and an electron for example) with wavefunctions $\psi _a (\vec{r}_1)$ and $\psi _b (\vec{r}_2)$ the combined wavefunction will be the product of these two

$$\psi(\vec{r}_1, \vec{r}_2)=\psi _a (\vec{r}_1) \psi _b (\vec{r}_2)$$

This is in accord with your intuition when you view the wavefunctions (actually their modulus squared) as a probablility density function. Where $| \psi _a (\vec{r}_1)| ^2$ gives the probablility (density) to find particle a at position $\vec{r}_1$ and $| \psi _b (\vec{r}_2)| ^2$ gives the probablility (density) to find particle a at position $\vec{r}_2$

$| \psi(\vec{r}_1, \vec{r}_2) |^2 = |\psi _a (\vec{r}_1)|^2 |\psi _b (\vec{r}_2)|^2[/tex] gives the probablility (density) to find particle a at position 1 and particle b at position 2. No surprises here, this is just the multiplication of probabalities. More interesting is to look at two indistinguishable particles a and b, two electrons for example. There is no way to distinguish one electron from another, they have no differing characteristics. In this case the combined wavefunction should be such that, and this is important, you can't tell which particle is in which state! The combined wavefunction can be either $$\psi _+ (\vec{r}_1, \vec{r}_2)=\psi _a (\vec{r}_1) \psi _b (\vec{r}_2)+ \psi _b (\vec{r}_1) \psi _a (\vec{r}_a)$$ or $$\psi _- (\vec{r}_1, \vec{r}_2)=\psi _a (\vec{r}_1) \psi _b (\vec{r}_2)-\psi _b (\vec{r}_1) \psi _a (\vec{r}_a)$$ (here I have omitted the normalisation constants) In both cases you can't tell if it's particle a at position 1 and b at two or the other way around, so quantum mechanics allows in this way for fundamentally indistinguishable particles. Particles that combine with the plus sign are called Bosons and those who combine with the minus sign are called Fermions. It turns out that Bosons always have half-integer spin and Fermions have integer spin. This can be proven, but it takes some advanced relativistic quantum mechanics, and even then there are still some people (Feynman for example) who doubt the validity of this proof. Now with this all in mnd the Pauli principle naturally follows. For two particles to be indistinguishable, they must have the same quantum numbers. In the case of Fermions the wavefunction $$\psi _- (\vec{r}_1, \vec{r}_2)=\psi _a (\vec{r}_1) \psi _b (\vec{r}_2)-\psi _b (\vec{r}_1) \psi _a (\vec{r}_a)$$ will be zero for [itex]\vec{r}_1=\vec{r}_2$. This means the probability two find two (the same) Fermions with the same quantum numbers at the same place is zero: The Pauli Exclusion Principle. The probability to find these two particles ever closer is ever smaller, resulting in a so-called 'exchange-force' between the particles.

Note that the probability to find two indistinguishable Bosons at the same place is 2 times as large as if they were distinguishable (if you like you can verify this yourself). This accounts for the fact that Bosons like to be near each other, which can be seen in a laser for example.

Maybe you already knew this and your question really was why do Fermions combine with the plus sign and Bosons with the minus sign, but to that there is, at an elementary level, yet no simple answer.

4. Jul 15, 2005

### TriTertButoxy

5. Jul 15, 2005

### da_willem

You're welcome, and that makes two of us. It is pretty humerous though...

6. Jul 15, 2005

### εllipse

I thought that a principle was an axiom and therefore wasn't derivable .. ? Is this incorrect?

7. Jul 15, 2005

### dextercioby

No. A principle is a postulate/ axiom.

Daniel.

8. Jul 15, 2005

### da_willem

I don't know if that holds in general, but the relation between spin and (anti)symmetry of the wavefunction is an axiom in nonrelativistic quantummechanics...

9. Jul 15, 2005

### dextercioby

Sort of. It's not in the axiomatical strucure for sure. In terms of nonrel. QM it is merely a SPECULATION.

Daniel.

10. Jul 15, 2005

Staff Emeritus
I don't understand your "SPECULATION" and your wink. Don't you believe the spin-statistics theorem holds for nonrel QM? In spite of Feynmann's demurral, it's well derived from the Wightman axioms in QFT.

11. Jul 16, 2005

### Nicky

Even in QFT, I thought that the commutation rules for fermion and boson fields were taken as axioms. The spin statistics theorem just explains why fermions are spin n+1/2 and bosons are spin n, not why fermions and bosons exist. Please correct me if I am mistaken, I've only just started learning QFT.

12. Jul 17, 2005

### dextercioby

Ok. I've said pretty clearly that in the terms of norelativistic quantum mechanics, ascribing a whatsover connection between spin & statistics would be a simple speculation, because there's no way one would begin with the 6-th postulate (the symmetrization one) and end up with the boson-->integer spin & fermion-->semiinteger spin.

Daniel.

13. Jul 17, 2005

### metacristi

Such questions show why the philosophy of science is indispensable for the scientific quest, the roots of science lie in philosophy. Science without philosophy of science is blind, despite its obvious pragmatic 'success' (which by no means amount to say with certitude that science approach us to the Truth).

Returning at your question yes you could see that postulate as a sort of axiom. However the great difference to the situation in mathematics is that it is always considered corrigible (in a nontrivial way going till its rejection from science if further research prove it sterile), that is accepted only on a provisional status. We are far from having some (known as) true first premises from which, together with a theory of good observations (empiricism), to answer (by deduction), step by step, all possible questions about Nature, the dream of Aristotle.

At the basis of scientific methodology a very important role plays the so called hypothetico-deductive method (deductive nomological): from a set of premises considered true (at least provisionally) we obtain a set of novel predictions (some of them corroborated/'confirmed' in the case of accepted scientific theories). But this does not imply that a scientific hypothesis should answer all the 'why' questions (implying causality) in order to qualify at the status of scientific theory. Not at all, a descriptive account/phenomenological account is enough.

So that in the set of premises are accepted also postulates (such as Einstein's postulates in SR and GR or Pauli's exclusion principle etc) as much as they are useful for the theory using them, helping the theory to give a more accurate description of the world at the observational level. The mechanism is like this: those postulates toghether with other accepted premises produce more (as many as possible) novel testable predictions at the observational level accesible to us + accomodate some known facts (at least some novel predictions confirmed experimentally).

The great observational success of GR for example fully justify the use of such postulates even if they (the postulates in the premises of GR) are not directly testable experimentally (testable in isolation). This fully entitle us to grant a fallible epistemological privilege to those postulates even if they are not required with necessity by observed facts (that is they are accepted provisionally as being part of science, accepted as corrigible). Pauli's Exclusion Principle proved fruitful and empirically evolving long ago-in 1928 Sommerfeld and Bloch explained using it the properties of electrons of conductions in metals, it explains the repartition of electrons in the Periodic System etc-it fully deserves its place in science at least currently. This notwithstanding that we deal with unobservables in the case of Pauli's principle, not amenable to direct empirical observations (we have only indirect ones at the direct observational level + confidence via the great coherence with the other (still) accepted parts of science; the fact that it is 'in line' with the standard formalism of QM does not amount per se to a direct confirmation, the standard formalism is at its turn provisionally accepted, and anyway gives only a descriptional view not a causal one).

Some go way further (such postulates are forever inside science being approximately true) but I don't think that the current situation enable such a stronger conclusion, both the foundationist and coherentist theories of truth/justification have (still) important problems; I'd argue that scientific knowledge is merely justified belief not justified true belief, fallibilism (in a non trivial way) should never be dropped especially in physics where the problems of underdetermination and theory ladenness are very important (in the light of the lessons of the history of science too).

Last edited: Jul 17, 2005
14. Jul 17, 2005

### mikeyork

No. Such an axiom isn't necessary. All you need to know is basic quantum mechanics -- in particular the necessity of choosing a unique state vector for a unique state description, if you want to compute interference effects -- plus a rule (a very obvious one) that says particle permutation is not observable (because particle ordering is an artifact of how the observer describes multi-particle states).

In particular, forget all the stuff that says that fermions and bosons obey a different type of quantum mechanics. Because that is putting the cart before the horse. Rather, it is the same QM that tells us they have a different statistical behavior in large ensembles. And forget all the stuff that says wavefunctions must be anti-symmetric for fermions. Because it isn't true, except in a special case (the usual, but not explicitly stated case) that a particular order-dependent (anti-symmetric) method of choosing the spin quantization frames is used.

In fact there is a single general exclusion rule that works for all particles, regardless of spin and which follows from the uniqueness and permutation unobservability I described above. It just happens that for spin half particles in a particular frame of reference it can be expressed as the Pauli principle. (An additional important factor in understanding this is that, for fermions, state vectors for state descriptions that are related by a 2pi rotation of their spin quantization frame differ by a minus sign. In other words to define unique state vectors, you must specify the relative orientation of the spin quantization frames. It is not sufficient to specify only the three axes in each case.)

A word of warning: Most physicists do not understand this yet. The conventional wisdom is still that either an additional axiom (The Symmetrization Postulate) is necessary (see da_willem's post, for example) or that it can be proved in Quantum Field Theory as a consequence of "(Relativistic) Local Causality". So if you try to argue the common sense description I have given without really understanding the math (see below) then you will get into deep water with people who (incorrectly) think they know better and to whom you won't be able to provide a convincing answer.
Here it is:

http://xxx.lanl.gov/abs/quant-ph/0006101

15. Jul 17, 2005

### mikeyork

So far you have given a good explanation of the conventional wisdom. As a follow-up to my last post, however, I'll point out a little problem with what you have written.

First off, you haven't defined the frame of reference for spin quantization. No matter, we'll assume you are using a canonical frame (the same frame in which you measure both position vectors) -- in which case the spin quantization frame is the same (in the sense of having the same axes) for both particles. However, since you are using 3-vectors instead of polar co-ordinates, we don't know the relative orientation of the two position vectors. If we do a 2pi rotation on one particle's frame of reference, but not the other, then we will change the wavefunctions for a half-integer spin particle by a factor -1, even though the frame axes end up to be the same. Using your notation, there is no way to distinguish the new wavefunctions from those that haven't been rotated. In other words, your single-particle wavefunctions are not unique (they are ambiguous up to a sign). Therefore, the sign of superposition will also be ambiguous since you cannot rule out the possibility of a 2pi rotation on one particle's spin quantization frame when you do the exchange. (Without uniquely defined relative orientations of frames, you don't have uniquely defined wavefunctions and so you don't have a unique definition of exchange.)

In fact it should be obvious that if you choose relative orientations of spin quantization frames so that one particle's frame gets rotated by 2pi by the exchange then your "anti-symmetric" wavefunction becomes a "symmetric" wavefunction in your ambiguous notation.

It turns out that when you set up a method for defining spin quantization frames in an order independent way then you can always get symmetric wavefunctions, regardless of spin. The conventional anti-symmetry is actually due to an implicit (but not obvious) geometrical asymmetry in choosing a canonical frame. (The details of this are in my paper referenced in my previous post.) Note that the observable properties are the same, regardless of which choice is made, because the inclusion or exclusion of an additional order-dependent phase cancels itself out when you examine the allowed physical quantum numbers that result from the interference due to the superposition.

Unfortunately, this conventional notion is convention dependent. As I have pointed out, one can always choose fermion states (obeying Fermi-Dirac statistics) that are superposed with a plus sign.

16. Jul 17, 2005

### ZapperZ

Staff Emeritus
This is because most physicists pay attention to work that has appeared in peer-reviewed journals. Scanning your paper, it appears that you cited yourself from papers you wrote that only appeared in the e-print archive. Can you please cite which peer-reviewed journals they have appeared in?

However, most importantly, can you show which experimental observations so far have proven that your theory is more valid than the "conventional wisdom", especially where there is an explicit effect due to a different "frame of reference of spin quantization"?

Zz.

17. Jul 17, 2005

### mikeyork

There was a crunch of space in the published conference proceedings. So I had to truncate either the paper or the references. I also had to leave out the appendix (that appears in the e-print). The first reference in that paper contains a comprehensive list of references to all papers using a similar (but incomplete) argument to mine published before 2000.
The conference paper was published in the proceedings. I see no point in publishing it again.

I made two earlier submissions to refereed journals that were rejected. Once because it was too long (it was indeed very long) and once because, according to the referee, "everyone knows" that fermion wavefunctions must be anti-symmetric, when the whole point of my paper was that this wasn't always true and isn't a good way to look at things.

After the 2000 conference I started a newer paper that would flesh out my conference paper and include more details on many-particle systems, intending to submit it to a refereed journal, but my job (in math finance theory) has kept me from finishing it.

I also think that forums such as this provide a much better opportunity for discussion than refereed journals. Referees can simply refuse to discuss the matter and often do. In a public forum such as this there is more incentive to engage in serious debate.

Indeed this is a lot more important. But for two-particle states, the claimed observable results are identical and cannot be used to discriminate. However, it is simply a matter of logic. The conventional view is self-contradictory since it claims a unique exchange phase for wavefunctions that are not themselves unique. This logical hole, was previously not noticed, or ignored. In fact I don't believe the 2pi rotation sign change for fermions was fully realized until some time after the symmetrization rules had become established wisdom. Everyone knew how to calculate in order to get the correct answer, so that is what they did. What I have done is to fill this hole.

For states of more than two identical particles, the observable rules for all such pairs are, again, the same. The critical area lies with composite bosons. Are they "true" bosons or does complete antisymmetrization of their constituents give different observable effects? This is a pretty complex issue, and the first questions about it date back to the 1920s and diatomic molecules. To my knowledge, all the data shows that they behave as "true" bosons as my theory predicts until the bosonic states are broken up by interaction. Plainly if the composite bosons are no longer in their identifying eigenstates (because of interaction) then my rules concerning composite eigenstates no longer hold. However, to my knowledge, scattering data such as alpha-alpha or d-d gives an unequivocal yes to my description of these composite systems as true bosons. I don't know what the anti-symmetrization argument predicts in terms of observable data, but as far as I can see, it should give different predictions (as I believe I have proved in my appendix). However, some people hold that it gives the same answer, which I do not understand and cannot verify as I have never seen the argument in sufficient detail.
I don't know what you are referring to here. Choice of spin quantization frame is arbitrary and has no observable effect. Any observable difference lies with states of three or more identical particles and differs according to whether or not complete anti-symmetrization holds (conventional view) or whether composite eigenstate rules obtain (my view) and whether there is a difference (as far as I can see there must be).

If you really want to get to the bottom of this, I suggest you do more than just "scan" my paper.

18. Jul 18, 2005

### ZapperZ

Staff Emeritus
So in other words, none of what you are claiming is part of mainstream physics AND has never appeared in a peer-reviewed journal. If that is the case, you may want to read our guidelines FIRST about "discussing" such things. This forum is unlike others.

This is VERY confusing. You are claiming that (i) there's no difference between yours and conventional theory and/or (ii) there is in some cases. Can you please make the EXACT citation to the particular experimental data that support your claim but not the conventional idea?

Other than that, I would request that, until you reread our forum guidelines, that you refrain from using your idea as an explanation to the issue at hand. You will understand that the validity of your idea has to be established FIRST, and that can only be done in peer-reviewed journals and by experts in the field. If you wish to use this forum to "work out" your idea, I would then refer you to the new Independent Research forum to submit your work. It no longer belongs in the main PF physics section.

Zz.

19. Jul 18, 2005

### metacristi

Unfortunately what you pointed out does not really offer the 'why' answer the OP was asking. You provide a formal description but this does not really amount to answer the 'why' question inside the standard formalism of QM, that is without interpretational elements (the assumptions made are questionable; others here tried entirely from inside the standard formalism but the solution offered is only descriptive not causal). As a digression here, descriptive formal explanations are distinct from explanations implying understanding, meaning either a normative/causal theoretical argument or a deduction from direct experience, not the case here for we deal with unobservables; anyway we have a big problem with the empirical base even at the level of observables, science 'is erected on a swamp', to quote Popper, everything is corrigible, we can only interpret experiments we do not deduce truths from observations.

The standard formalism of QM, the only one which really makes testable predictions, is descriptive in nature, none of its proposed ontologies, interpretations, really have the edge, we deal with a strong underdetermination here, not to mention theory ladenness (this despite the arguments of many leading physicists-some of them hinting a very poor understanding of modern philosophy of science-that only the current main views are acceptable the rest being 'dead ends' forever).

Feynman mentions that Pauli tried to base his principle on QFT and Relativity but this is questionable. So its safer to follow the advice of the same Feynman (in the 3rd volume of his lectures) that the anti symmetry of the wavefunction(s) for fermions (and subsequently Pauli's principle) is a fundamental principle of how nature works.

I'd argue that in this state of research, that is for the moment at least, we have to be contended to consider Pauli's assumption a fundamental principle of nature, provisionally accepted inside sicence. Which, of course, by no means amounts to say that this will ever be the case, maybe we could deduce it, from 'deeper' assumptions by showing that one of the intepretations is really superior. What if it would be a hidden variables approach, realistic in nature, even if we will have to drop the invariance of laws in all frames, after all Brans-Dicke hypothesis is a serious contender to GR even now...

Last edited: Jul 18, 2005
20. Jul 18, 2005

### reilly

mikeyork -- Virtually all of chemistry depends critically on the standard anti-symmetric electron wavefunctions, not to mention superconductivity, and particle theory, and most everything else. Do you find different results in atomic or any physics? -- relative to the standard approach.

Regards,
Reilly Atkinson