One has probably first to answer a question what is a correct definition of a quantum system within which the Pauli Exclusion Principle applies. Two seperate atoms certainly contain electrons not subject to the common exclusion principle. In a big quantum system with a continuum of quantum states there is practically no Pauli exlusion effect. So the distance does not seem to be the most important parameter but rather range of quantum coherence.

The Pauli exclusion principle can be expressed by saying that the wave

function of a system of identical fermions must change sign when you

exchange two particles. Stated in this way, it does not cease to be

valid with the distance. I elaborate a bit on your example of quantum wells in the attached document.

@Giancarlo: If the particles are identical, that is they are indistinguishable, then we must have |psi(r1,r2)|^2 = |psi(r2,r1)|^2, where psi(.,.) is any solution of the Schroedinger equation for exactly two particles, interacting or not (there is no world apart from those two particles). Now, if psi(.,.) is a real function, then we have exactly two possibilities: either psi changes its sign after the particles are exchanged, or it stays unchanged. Particles obeying the first case are called "fermions" and the other ones "bosons". Not the reverse way, as you try to say in the first sentence of your reply. Additionally, what you have written in an enclosed material is aesthetically pleasing but, unfortunately, meaningless when it comes to two potential wells not deep enough to have more than one bound state each. Neverthelss, experimentalists keep saying that two electrons in nearby quantum wells repel each other, just because of Pauli principle at work. This effect is not observed for better separated wells, so the effective interaction must be either short-ranged or quickly decaying with distance. Hence my question.

@Marek: I'm not sure to understand your first comment: I'm exactly saying that psi changes sign when the fermions are exchanged.

Concerning your second comment, I apologize because I did not fully appreciate the spirit of your question. Now it is more clear to me. I think that in the case you are mentioning (just a single bound state) we should construct the fundamental level of the two fermion system by using the single well bound state and unbound states (which are in the continuum). I think that in this case the fundamental level of the system will not be a bound one. However I also expect that the probability of finding one of the two particles far from the wells should become small when the wells are quite far away. I'll try to check this with a more definite model: this should give a more quantitative answer to your question.

@Marek: after a second tought (and some calculations...) I think I can give you a more clear exposition of what I think is the point. Let us suppose that we have a very simple potential, for example two Dirac delta wells separated by a distance d,

V(x) = -k Diracdelta(x-d/2) - k Diracdelta(x+d/2) with k>0

Now, Diracdelta(x) has just a single bound state, let us call its wavefunction G(x).

If d is very large as I said in my previous answer you can construct the ground state for V(x) using the symmetric combination G(x+d/2)+G(x-d/2) and the first excited one by the antisymmetric combination G(x+d/2)-G(x-d/2).

In this regime the two states are almost degenerate, the energy splitting being proportional to the overlap between the exponential tails of G(x+d/2) and G(x-d/2).

What is more important, the two states constructed in this way are both bound states, and you are exactly in the situation described in my previous attachment: both fermions can be bound, and the Pauli principle manifest itself by depressing the probability that both are near the same well. If you want you can look at this as the effect of an effective repulsion which never disappear.

When d decreases the overlap between G(x+d/2) and G(x-d/2) increases, and at some point the antisymmetric bound state cannot exist anymore. If I did the calculations correctly this should happen when d

@Giancarlo: Thank you for the excellent reply. It is very refreshing to see that abstract Hilbert space having nothing in common with ordinary geometry can be rigorously "translated" into something more "touchable". You did a great job. In my opinion it well deserves publishing, maybe in Phys. Rev. Letters? It's not a breakthrough, but very interesting result per se. Please accept my sincere congratulations!

@Marek: Thank you for your kind words.

It would seem that experimenal info bearing on this question is sparce. The only physics I can think of that does is perhaps the collapse of a neutron star when it receives matter taking it over the weight limit for producinng a black hole. If we receive that as really happening, it may say that Pauli exclusion works against all 3 strong forces but not against gravity. Since in a neutron star, exclusion does support against collapse in densities not far off nuclear densities, distance would not seems to be the operative variable but mass seems likely. Probably there are a plethora of theories that bear on this and give different answers.

Giancarlo Cella and Adam Jacholkowski have together given a sufficiently complete answer the to question, at least in lab systems on earth. Dr. Cella's calculation for a simple two-well geometry shows that, when the two wells are "close together" (that is, close enough for a non-negligible probability that a particle bound in one well could tunnel across the potential barrier into the other well) the existence of the second well changes the structure of the energy levels in the first well, and vice-versa. It is exactly in these kinds of cases where the Pauli Exclusion Principle plays an important role, since two particles in the two wells overlap sufficiently that their wave-functions interact, and they cannot occupy the same quantum state (if they are fermions). As Dr. Cella indicates, when you move these two potential wells further apart, the separate energy levels created by the interaction of the two wells begin to merge (become degenerate)-- for physical potential wells (as opposed to idealized Dirac Delta functions) non-degenerate energy levels in each potential well should converge exponentially toward degeneracy as a function of distance, when the two wells are far enough apart to be in the "cutoff region" (non-propagating region) for the particles. At some distance this exponential cutoff means that the Pauli Exclusion Principle, though theoretically still operating, has negligible (exponentially small) effect.

Jacholkowski's comment summarizes this example calculation and all cases like it in a short phrase: "the distance does not seem to be the most important parameter but rather range of quantum coherence" -- so the "range" of the Pauli Exclusion Principle is theoretically unlimited, just like gravity, nuclear, and electromagnetic forces, but the EFFECTIVE range of each force is determined by the mathematical details of the interaction in a particular geometry-- in this case, the geometric and energetic details of the potential wells, and the mass and energy of the particles trapped in the wells, determine the quantum coherence length between two wells and the distance at which they effectively decouple and the Pauli Effect becomes negligible.

It is a little bit harder for me to understand the definition of the coherence length in a "white dwarf" star-- a star which is no longer held up by thermal gas/plasma pressure nor by radiation pressure from fusion, but which resists collapse into a neutron star because the "Fermi Sea" of electrons in the plasma resists collapse into a smaller volume due to Pauli Exclusion. I am going to suppose that the coherence length there is set roughly by the size of each atom, the 10^56 atoms in a typical white dwarf star forming a kind of random (gas-like) "lattice" of moving potential wells, and that when gravitational force exceeds the weak nuclear force at the core of the white dwarf, that compresses the electrons and nuclei close enough together that Electron Capture takes place in the protons and initiates the Nova collapse into a neutron star. Somebody with white-dwarf knowledge out there, can you comment on the Pauli Exclusion "coherence length" typical in a white dwarf star?

No two same particles occupy the same space. Simple. For further explanation consult you local quantum chemist.

With all due respect, Dr. Baginski, I think you have oversimplified. Two identical Bosons can, and often do, occupy the same space. But we are talking about Fermions and, for all practical purposes, we are "your local quantum chemist" and we are discussing the details of that "further explanation"

Thank you for the clarification.

Not apparently for the neutron star system.

I would like to add to Vesselin's completely correct answer, that quantum mechanical computations concerning spectral lines of atoms are all based on the unrestricted validity of Pauli's principle and thus provide a huge ammount of evidence for the correctness of this unrestricted form.

This is an instructive situation since the values of multi-particle wave functions are not observable quantities. So, whether they actually behave according to Pauli cannot be tested by inspection. Therefore, verification can happen only indirectly, for instance through correct predictions of spectroscopic data.

Also, it is not at all clear how a modification of Pauli's principle depending on particle distances could look like. What seems to be clear is that such a modification would also be a considerable complication.

Vesselin and Ulrich,

Of course, you both are right and I do not intend to question your categorical statements. Perhaps my original question was formed not clearly enough, but in reality I was interested in a situation nicely worked out by Giancarlo Cella. He presented quite quantitative answer, not fuzzy like the one given by Vesselin in his contribution (see his last statement).

Electronic energy bands may be obtained just by solving *single* electron Schroedinger equation with periodic potential in reciprocal space, without any direct resort to Pauli principle. But, indeed, one can think about the occupation of many, many available energy levels as being ruled by Pauli principle (and temperature, of course). Maybe this is why metal pieces do not explode. Thank you, Vesselin, for mentioning electrons in metals.

So, everything should be clear and fine. Except, maybe, for the neutron stars, mentioned by James Langworthy. I hope the idea put forward by Giancarlo Cella is applicable to them as well.

Unfortunately, our friend Vesko (Vesselin) Dimitrov is no longer with us:

http://www.legacy.com/obituaries/idahostatejournal/obituary.aspx?n=vesselin-dimitrov&pid=161352699

I'm a sorry to be the bearer of this tragic news.

There are already many good answers, and a detailed one from Giancarlo Cella. But if you ask me a simple answer is based on overlap of wavefunctions that calls in Pauli exclusion principle. As far free electrons in metals, the Hartree-Fock theory is based upon antisymmetric wave functions and Pauli exclusion principle is very well a part of it.

Dear Marek,

I think that this an important but bad question. Important because the boundaries between classical and quantum are badly delineated and usually the intuition is feeble in this regime, bad because certain things has to be taken literally. So if you take literally Heisenberg's uncertainty principle (with whatever it means; measurements or dual space relationships: Fourier transform) it does define "quantum cell" so to speak. This cell is very elastic and if one dimension goes large another goes small so it is a "volume" therefore a cell. If one sees it this way then quantum means something that it is in a cell. Because a single dimension of a cell can be theoretically infinite any quantum phenomena must by definition be everywhere. We do not see it because usually there are limitations imposed on other dimensions and we almost never reach large size of quantum dimensions. This has everything to do with quantum isolated state. However, experiments with quantum nonlocality (whether the photons or electrons) prove that large space separation conserve entaglement on very large distances. Therefore the Pauli exclusion must be by definition applied to entanglement (that defines quantum states) and cannot have size limitations. The legitimate question would be rather, under what set of conditions we can see it as working on very large distances or what is the size of customary fluctuation that kills it. So in essence late Vesko already rendered the verdict. I only felt compelled to tight in loose end. Math oriented people not always see it the way I do.

As a last note; Sanjay show some patience. John is present on many threads and not always people are calling him what you called him. John I would seriously recommend at least make your equations clear so they do not raise factual objections. As I see them (probably in loose translation of Tech onto the java used at this site) they do not make much sense.

Let us consider an ensemble of fermions that are described by a complete set of commuting observables. The exclusion principle applies only when all the quantum numbers associated with this CSoCO, except those to do with spin, are the same. There is a very good discussion in Claude Cohen-Tannoudji et al's two volume set "quantum mechanics" about this subject.

Dear Giancarlo,

The confirmation of the accuracy of your calculation can be found in the following paper posted on my Research Gate page:

On the spectrum of the Schrödinger Hamiltonian with a particular configuration of three one-dimensional point interactions

Silvestro Fassari, Fabio Rinaldi

Reports on Mathematical Physics (impact factor: 0.64). 12/2009; 64:367-393.

There is only one possible answer to that question: The Pauli principle is ALWAYS valid. It does not depends on the distance, neither in any other propertie. The only thing important here is that electrons are fermions, and many-fermions wave functions MUST change their sign upon the exchange of any two fermions.

Pauli's exclusion principle is violated in extreme conditions- for one- those arise in neutron stars. Atomic structure is demolished and electrons combine with protons to form neutrons. Pressure and densities are very high- one can figure out the distance at which it happens- much less than atomic scales.

Neutron star can be seen as a macroscopic nucleus except we have a very large density of quantum states, so I am not sure Pauli principle is really violated in this case.

Theoretical conjectures can be argued forever. What we know is that when a neutron star reaches a certain mass, it collapses into a black hole. Since Pauli exclusion has been the only thing keeping it from collapsing at lower masses, it would seem Pauli exclusion fails. This doesn't prove it but until someone offers a mechanism to save the Pauli exclusion, all we have is the bare observation.

to distance up to which the coherence of quantum mechanics remains valid in a fluctuating anvironment. In a zero temperature universe (with background radiation at absolute null temperature) the Pauli exclusion principle will remains valid up to infinity.

Article Can fluctuating quantum states acquire the classical behavio...

At very high temperature (with mean particles energies at relativistic values (as in very intense gravitational fields)) the quantum coherence distance can become smaller than the distance of elemental particles (basically they becomes classical-like free) and hence the Pauli principles can be violated. At a temperature of 10**9 °K the quantum coherence distance becomes of order of 10**-10 cm for a particle owing an elctron mass. Basically we agree, don't you?.

@James (& Sanjay) ... is the use of the word "know" (by James) valid in connection with something never experimentally observed (gravitational collapse)? Or did you really mean just "current theory predicts?" I believe everyone "thinks" (not "knows") there is likely to be some resolution to this other than the singularity. Besides that it is theoretically unobservable, we have yet to examine BH's at reasonable resolution and distance to confirm even the externally verifiable parts of GR.

On a related line of thought, I have often wondered about Pauli exclusion and black holes, and have read something about calculating the breakdown pressure that I cannot now find. If anyone can locate it, please post on this thread.

But as I think of it now, I see no reason gravitational or other pressure cannot simply reduce the spatial separation associated with energy states so that in the limit it approaches zero (kind of like renormalization).

Then again, exclusion applies only to fermions. A large part of the mass of complex fermions (e.g. protons) is tied up with the component bosons (gluons) and the bound kinetic energy of the component fermions (quarks). How does that affect the question?

If two fermions with different energy are absorbed by a black hole, is the end state different? If it is the same, what process makes it the same and where does the delta energy go?

True, I was assuming we had observations of a neutron star collapse by accretion from a binary partner but that seems not yet to be observed. What we do know is that stars over a certain mass do collapse when their sustaining fusion reactions run out into black holes. The remnants have been observed.

Thanks, Sanjay, for your detailed response. I had never read an account of relativistic degeneracy and its role in collapse.

If these calculations are done in the reference frame of a particle involved in the collapse, the results are straightforward but are postponed infinitely far into the future from an external observer's point of view, an inconvenient distortion. If the calculations are done in a remote observer's frame, as I have pointed out in two papers, the slowing particle velocity due to time dilation should be associated with a mass increase, in order to conserve momentum.

Then at some point as the star's radius approaches the Schwarzschild limit, the implicit inertial confinement due to time dilation would begin to exceed, it seems to me offhand, the rate of relativistic degeneracy, possibly relieving the pressure? Have you ever seen a discussion of this? If in fact stars sort of froze into equilibrium just prior to contracting behind an event horizon, would there be any difference implicated in our astronomical observations? They would still be incredibly dark.

I have been read that QCD explains the quantum states of the Omega minus and the Delta++ baryons by adding a new degree of freedom to the Pauli exclusion principle. In so doing it ignores the existence of multiple configurations inside the proton that explain these particles’ state by ordinary quantum mechanical laws. However, the fact that the Proton Spin Crisis was not resolved, and that this crisis can be resolved immediately by using multiple configurations, make the whole point of view of the community questionable.

Of course at enough high temperature and density hadrons can melt into a state we call quark-gluon plasma in which the internal degrees of freedom show up, so the Pauli principle has to be applied to quarks only, while gluons as bosons escape from this constraint. One can suspect that in the core of the dense neutron stars it may happen, converting some fraction of the original up and down quarks into strange quarks which may be energetically favorable given the Pauli exclusion principle.

Dear Professor Adam Jacholkowski

Really, you explanation is very nice, I am agreed with you nice opinion.

thank you for your kind consideration.

Interesting discussion, but there are some caveats in the latter stages :

Sanjay Sood : "But for a very massive star (mass greater than 25 solar masses) this is not the end. These stars continue to collapse beyond the neutron star stage and now the velocities of individual neutrons reach the relativistic domain, thereby giving rise to a relativistically degenerate gas of neutrons. At this point the degeneracy pressure of the exclusion principle is no longer sufficient to prevent further collapse, in other words Pauli's exclusion principle breaks down completely. The star goes into an endless collapse, eventually disappearing entirely from our visible universe."

That is a (semi-)classical argument, and although we all agree something like this needs to take place, the quantum aspects of this mechanism are unknown, since we really do not understand quantum gravity! Neutrons are of course no longer the correct degrees of freedom for this domain.

As Adam says above, QCD is not a way out. There is active research into quark stars, and there are cogent arguments that the heaviest neutron stars are likely to have free-quark cores (typically, in one of the wonderful sueprconducting phases called "Colour-flavour locking" or even in a LOFF state).

The Pauli principle is quantum-mechanical, and I don't believe this can be violated; what can happen is that we get a unification between gravity and QCD, and that thus at the highest pressure we convert neutrons->quarks->to other degrees of freedom (bosonic? Higgs particle? ....) which can condense and thus the Pauli principle is still perfectly correct.

Of course heat can play a role as well. The arguments above are mainly for cold objects (neutron stars are very cold on a nuclear scale), and as said in the early answers, heating fermions leads to a mixed state that behaves almost classically, and thus the exclusion does not apply--it strictly only applies to pure states.

Hi Sanjay, regarding your question about what I meant . . . sorry I wasn't clear. It is a bit complicated, and after reading your question I realize there are several possible cases. Just before an event horizon forms, time dilation at the radius where it is going to form approaches infinity. There is a paradox as to how the event horizon ever forms. At the time of my question above, I was only thinking of how this played out with relativistic degeneracy, but here on a fresh day I don't see why I should limit my doubts to that case.

Mostly relativists do not write a lot that is understandable about how the formation occurs. Only after. I do not manipulate the full GR equations myself, though the Schwarzschild formulas for time dilation and so forth are simple enough. Once the event horizon has formed, I'm sort of reduced to reading articles written by those who do. And they vary. Sorry I cannot now provide links. But some refer to the infinite slow down at the horizon you mention, and some seem to suggest that even in the frame of observer 1 (the external one) the in-falling object disappears after a brief time (possibly depending on size, but a second or less). I cannot really evaluate this for myself or argue one side or the other.

But since this discussion thread has rambled into the subject of the collapse, and the process of going from no event horizon to having one, I feel slightly more comfortable asserting a chain of causality. As the formation of an event horizon gets close, time dilation approaches infinity. All physical processes slow down. It is not obvious the horizon can ever form.

Before you invoke proper time again (i.e. frame of observer 2 in-falling participant in the collapse), I will argue in anticipation of that. Before the horizon has actually formed, every time point for observer 2 can be transformed to a time point for observer 1. But observer 1 has no time points that include an event horizon because of the asymptotic slow down of physical processes where it would have formed.

Hi Sanjay, Interesting analysis, but actually I was talking about the formation of the event horizon rather than the singularity. Look at the moment just before any event horizon exists. There will be a lot of time dilation. As we get closer in time to the formation of an event horizon, then in the vicinity of the location where that event horizon will form, time dilation approaches infinity. As the formation of the event horizon may depend on continued collapse, especially for objects just barely massive enough to form an event horizon, then the time dilation could potentially prevent formation of an event horizon in the reference frame of an observer. Then there would no event horizon ever really existing in that observer's reference frame.

It might be possible to extend this argument over a broader range of masses. Perhaps no matter the total mass, prior to the event horizon forming, some of the required mass must be outside the radius where it will form? In that case, there might really be no black holes in our universe. Only objects which will in infinite time become black holes.