Navigating Wikipedia for Scientific Answers: Tips and Tricks for Laymen

[Ken G]

For a Bose gas of conserved particle number, and for a Fermi gas, the equation of state $P = \frac{2}{3}\frac{E}{V}$ still holds. For a photon gas, where particle number isn't conserved, we instead have $P = \frac{1}{3}\frac{E}{V}$.

I don't want to seem picky, but the issue there is not whether the particles are conserved or not, it is how relativistic they are. Relativistic particles of all types have the 1/3 coefficient there, non-relativistic have the 2/3. But your main point is well taken-- pressure just comes from kinetic energy density, so if you want to understand what the pressure is, you have to understand the energy history. Too much can be made of the PEP in this context, it just depends on how you are tracking that energy history whether you care about the PEP at all.

 

[Jabbu]

In any event, the magnetic properties of the neutron have little or nothing to do with the pressure inside a neutron star.

The OP question is about Pauli exclusion principle being a consequence of some fundamental force, or not. I thought it is generally believed that every interaction is at its essence a consequence of one of the four fundamental forces, even though they can be expressed in different theories in terms of probabilities, curvatures, or virtual particles.

You don't think the electrons filling the shells of a noble gas atom have important magnetic effects do you?

I don't know. Some say it is impossible to separate electric and magnetic effects.

The PEP appears in how differently those electrons act if they have the same or opposite spins, and that's way more important than any magnetic effects.

The question is only where it is derived from and what it is based on.

http://en.wikipedia.org/wiki/Angular_momentum_coupling

I see here spin-orbit and spin-spin coupling are described as magnetic interactions. Is spin-spin coupling not the same effect PEP is talking about?


http://en.wikipedia.org/wiki/Magnetism

And when this articles talks about relation between magnetism and mathematical formulation of quantum mechanics, in particular regarding spin and the Pauli principle, than what does it really mean and where can we see this "mathematical formulation"?

 

[Ken G]

The OP question is about Pauli exclusion principle being a consequence of some fundamental force, or not. I thought it is generally believed that every interaction is at its essence a consequence of one of the four fundamental forces, even though they can be expressed in different theories in terms of probabilities, curvatures, or virtual particles.

It all depends on what you mean by "an interaction". If I have a hot gas in a box, and I open a door to the box, the gas comes shooting out. In a fluid description, we will say the gas accelerated out of the box. But there will not be any forces on that gas, so no "fundamental interactions" are involved in the acceleration of that gas-- it's just a bunch of particles that were carrying momentum and that's what they do when you open the door. This is the nature of pressure, so when one is talking about degeneracy pressure, this is what one is talking about-- not fundamental interactions.

I see here spin-orbit and spin-spin coupling are described as magnetic interactions. Is spin-spin coupling not the same effect PEP is talking about?

No, the PEP is talking about statistical mechanics, not forces between particles. The only type of force that the PEP directly gives you is the type you get when you look at a whole system and see how the internal kinetic energy E changes with volume V, so dE/dV. That is what we mean by pressure, but it's not a force between particles, it is a hypothetical change in energy of the particles when you hypothetically change V. You could effect that change in V using any fundamental force you like and it will make no difference-- you end up with the same dE/dV because that's a property of the system that has nothing to do with forces between the particles.

And when this articles talks about relation between magnetism and mathematical formulation of quantum mechanics, in particular regarding spin and the Pauli principle, than what does it really mean and where can we see this "mathematical formulation"?

It involves the entire study of quantum mechanics, I'm not sure there is a shortcut. The key point is, in physics it's important to know when you need to worry about certain effects, like magnetic forces, and when you don't need to worry about them, like when you are using the PEP to understand the statistical properties of a bunch of fermions at very low temperature.

 

[tom.stoer]

The OP question is about Pauli exclusion principle being a consequence of some fundamental force.

I think I presented an argument that the Pauli exclusion principle has nothing to do with any force or interaction; it is valid even for free particles w/o any interaction at all.

Introducing fermionic creation operators an electron with quantum numbers denoted by 'r' is described by a fock space state

|r\rangle = a_r^\dagger|0>

where $|0>$ is the vacuum state. All states can be constructed that way. And all interactions can be expressed in terms of these creation (and annihilation) operators.

Now because of the Pauli principle (or the more fundamental spin-statistics theorem) we construct the creation operators such that the following holds

a_r^\dagger\,a_s^\dagger = -a_s^\dagger\,a_r^\dagger

For $r \neq s$ this means that a state with two different fermions is anti-symmetric w.r.t. to interchange of to particles.

But for $r = s$ this means

(a_s^\dagger)^2 = 0

So we construct the mathematical formalism such that no two identical fermions can be described by the formalism

The spin-statistics-theorem demands to quantize spin-1/2 fields using the fermionic i.e. anti-commuting field operators. These operators guarantuee purely algebraically that the Pauli exclusion principle is satisfied. Everything else like interactions, atomic orbitals, pressure, macroscopic effects, ... comes on top of or can be derived from these fundamental rules.

 

[Jabbu]

The spin-statistics-theorem derived from CPT demands to quantize spin-1/2 fields using the fermionic i.e. anti-commuting field operators. These operators guarantuee - as described above - that the Pauli exclusion principle is satisfied. Everything else like interactions, atomic orbitals, pressure, macroscopic effects, ... can be derived from these fundamental rules.

Rules and statistical simplifications can not be more fundamental than physical properties which define them. "Pauli's Exclusion Principle: The Origin and Validation of a Scientific Principle" by Michela Massimi, explains how spin and the Pauli principle were directly inferred from experiments specifically related to magnetic fields, and how spin was incorporated into Schrodinger's equation based on magnetic moment. Spin and magnetic dipole moment are intrinsically related, if not just different names for one and the same thing.

 

[Nugatory]

Spin and magnetic dipole moment are intrinsically related, if not just different names for one and the same thing.

A charged particle with non-zero spin will have a magnetic dipole moment, so yes, the two concepts are related. But claiming (as you did above) that they are just different names for the same thing is as absurd as claiming that because some domesticated dogs wear collars, "domestication" is just another name for "collar".

 

[tom.stoer]

Rules and statistical simplifications can not be more fundamental than physical properties which define them.

The are no statistical simplifications. And I do not see what the "physical properties" are and what they define. Please explain.

RulesPauli's Exclusion Principle: The Origin and Validation of a Scientific Principle" by Michela Massimi, explains how spin and the Pauli principle were directly inferred from experiments specifically related to magnetic fields, and how spin was incorporated into Schrodinger's equation based on magnetic moment.

I agree to the historical original. But the fact that a fundamental theory is derived (induced) from experiments and phenomena does not mean that these phenomena are more fundamental than the theory.

Spin and magnetic dipole moment are intrinsically related, if not just different names for one and the same thing.

The magnetic moment can be related to spin, but spin is not necessarily related to a magnetic moment. There are elementary particles with non-zero spin but zero magnetic moment. Spin can be described (defined) w/o ever referring to magnetism, charge or anything like that.

 

[Ken G]

What's more, integer-spin particles can have magnetic moments, and do have spin, but do not satisfy the Pauli exclusion principle, so apparently that principle must be something a bit different from having a magnetic moment. But of course the mind needs to be open to learn anything.

 

[Jabbu]

A charged particle with non-zero spin will have a magnetic dipole moment, so yes, the two concepts are related. But claiming (as you did above) that they are just different names for the same thing is as absurd as claiming that because some domesticated dogs wear collars, "domestication" is just another name for "collar".

When Wikipedia redirects "electron spin" to "magnetic dipole moment", when no article makes any distinction between the two, and when the terms are used and can be read interchangeably while the sentence retains the same meaning, then it's pretty difficult to differentiate the two.

Can you point any article on the internet that makes some distinction and somehow separates spin and magnetic dipole moment into separate entities or properties? What's the difference?

 

[WannabeNewton]

Is an exchange interaction similar to two potential wells that are next to each other, that gain just enough momentum to exchange?

I don't know what it means for potential wells to "gain momentum". Anyways, see here: http://en.wikipedia.org/wiki/Exchange_interaction

But as Ken mentioned, the PEP and the associated exchange interaction, while of course strongly related, are not exactly the same thing.

 

[tom.stoer]

Can you point any article on the internet that makes some distinction and somehow separates spin and magnetic dipole moment into separate entities or properties? What's the difference?

Spin, spinors, spin manifolds etc. are mathematical entities w/o any relation to interactions, charge, current etc. They are more related to geometry of spacetime.

http://en.m.wikipedia.org/wiki/Spin_structure
http://en.m.wikipedia.org/wiki/Spinor

Spin magnetic moment is defined in terms of spin and charge.

http://en.m.wikipedia.org/wiki/Spin_(physics)#Magnetic_moments

 

[Nugatory]

When Wikipedia redirects "electron spin" to "magnetic dipole moment", when no article makes any distinction between the two, and when the terms are used and can be read interchangeably while the sentence retains the same meaning, then it's pretty difficult to differentiate the two.

You're discovering that learning physics from wikipedia doesn't work; wikipedia just isn't good enough at bringing out the relationships between concepts. Here you would have been better served if "electron spin" had redirected to http://en.wikipedia.org/wiki/Spin_(physics).

 

[stevendaryl]

As Wannabe Newton (or somebody) says, the appearance of repulsion between identical fermions is a consequence of Fermi statistics: the fact that any two-particle fermion wave function must be antisymmetric under exchange of the two particles. That is, \psi(x_1, x_2) = - \psi(x_2, x_1). It's just a provable fact that if there is no degeneracy (that is, if different states have different energies) then an antisymmetric two-particle state will have a higher energy than a symmetric two-particle state, because it's not possible for both particles to be in the same, lowest energy level. So it's not really any kind of force. The energy cost for being anti-symmetric depends on the situation. There may be no extra cost if there is degeneracy (multiple one-particle states with the same energy level).

The flip side of Fermi statistics is Bose statistics: For two Bosons, the two-particle wave function must be symmetric under exchange of two particles: \psi(x_1, x_2) = + \psi(x_2, x_1). To me, Bose statistics are not mysterious at all. Since the particles are identical, switching them can't possibly make any difference. Fermi statistics is a little weird, because switching two particles does make a difference--it changes the sign of the wave function. However, the overall phase of the wave function is unobservable, so this doesn't count as a physically meaningful change.

 

[HeavyMetal]

I don't know what it means for potential wells to "gain momentum". Anyways, see here: http://en.wikipedia.org/wiki/Exchange_interaction

But as Ken mentioned, the PEP and the associated exchange interaction, while of course strongly related, are not exactly the same thing.

Hahaha, I was pretty drunk last night. Sorry, let me rephrase:

If we have two particles in side-by-side potential wells, is an exchange interaction somewhat of a physical "switching" between these two particles (even if they are identical particles)? I know we can't be sure because there is nothing distinguishing the two particles, but can it be thought of this way?

Or am I way off my mark, and it is literally just a mathematical consequence of the indistinguishability of the two particles? I guess exchange follows from the fact that you must have a symmetric and antisymmetric combination, and therefore it also means that switching the sign may or may not evoke a change in the wavefunction. Thanks :)

 

[WannabeNewton]

AFAIK it's just a mathematical consequence. There is no literal or physical switching of the particles involved.

This mathematical abstraction directly leads to a difference in the enumeration of states in a thermal ensemble of fermions vs. bosons when calculating the partition function $Z$. So the rather abstract notion of symmetric vs. antisymmetric states upon exchange directly affects the (also abstract) statistics of the system depending on whether it is a system of fermions or of bosons, because as you may know $Z$ determines the statistical problem entirely. And from $Z$ we get physical quantities like the pressure $P = \frac{1}{\beta}\frac{\partial}{\partial V} \ln Z$.

 

[HeavyMetal]

Thanks bud, you're awesome. You always go above and beyond in your posts and I learn from it!

 

[Ken G]

The way I think of the exchange interaction is, there are two orthogonal ways, symmetric and antisymmetric, for writing the state of a system of two particles such that no observation can tell which particle is which. Which choice you make of course corresponds to bosons and fermions, but the key point is, you will get a different answer for the interaction energy of the two particles (if they have a mutual potential energy based on some interparticle force that is normally neglected in degeneracy pressure calculations), based on which choice you make. Thus there are two fundamentally different ways, bosonic and fermionic, that indistinguishable particles can interact with each other via some interparticle potential energy function. The act of swapping them does not change that energy, the change comes from the fact that the interaction energy is different in the two ways you can assert the swapping requirement. So we should really talk about the "fermionic interaction energy" and the "bosonic interaction energy", which are both in contrast to the absence of any interaction energy when the particles are distinguishable and no swapping principle is in play.

 

[Peter Nordquist]

Neither electrons in a white dwarf, nor neutrons in a neutron star, "repel each other" in the sense of a force between the particles (the electron-electron electric forces are highly shielded by protons). The only sense in which there is a repulsion is the same way that ping pong balls bouncing around inside a box will "push on" the walls of the box-- they have kinetic energy, and that energy increases if you contract the box, so there is a dE/dV where E is the kinetic energy and V is the volume of the box. -dE/dV is what we call pressure, and it is there because the particles carry momentum, not because they repel each other. Exactly the same is true of the PEP for either electrons or neutrons. So that's why I say there is only a "force" in a globally averaged sense, accounting for the way the particles carry momentum around, not in the reductionist sense of how individual particles are interacting with each other.

Indeed, the PEP is playing no role at all in the pressure, once you specify the energy situation. All the PEP does is make it impossible to remove heat, which has a significant impact on the energetics.

Would this answer explain “How all those “electrons-repelling-each-other” clump together in the clouds in such massive numbers as to allow lightning to strike?” Is Coulomb's Law a generalization that is refined by quantum mechanics, meaning that electrons don't repel each other all the time? I'm thinking of an article I read on quantum dots that explains that electrons arrange themselves in quantum states inside the confines of the quantum dot inspite of there being no nucleus of an atom to center on. [First reply of mine outside the new person forum. Thanks for your consideration]

 

[stevendaryl]

Would this answer explain “How all those “electrons-repelling-each-other” clump together in the clouds in such massive numbers as to allow lightning to strike?” Is Coulomb's Law a generalization that is refined by quantum mechanics, meaning that electrons don't repel each other all the time? I'm thinking of an article I read on quantum dots that explains that electrons arrange themselves in quantum states inside the confines of the quantum dot inspite of there being no nucleus of an atom to center on. [First reply of mine outside the new person forum. Thanks for your consideration]

No, that's not quite right. Electrons really do repel each other through Coulomb forces. It's just that quantum effects (the Pauli exclusion principle) go beyond that repulsion in making sure that no two electrons occupy the same space.

The Pauli exclusion principle does not play much of a role in the lightning produced in a cloud, as far as I know.

 

[Peter Nordquist]

Interesting . . .and after reading your reply just now, I searched on "quantum theory of lightning" and noticed these two articles that came to the top of the search:
Record energies force new thoughts on lightning
http://www.physicscentral.com/explore/action/lightningforce.cfm
[where we read]
Most pilots avoid flying through thunderstorms, but it's now worried that if you were flying past one of these storms in an airplane, you could be exposed to dangerous radiation. In fact, scientists say that the energy some lightning can produce would be impressive if it were coming from an eruption on the surface of the sun.

What makes the mystery even greater is that lightning strikes millions of times every day, but only a very small fraction of those storms can produce Terrestrial Gamma Ray Flares (TGFs) . It's unclear to physicists how certain storms can create them when others don't at all.

Recently, another major discovery was also made by NASA's FERMI spacecraft . Scientists used the orbiter to glimpse large amounts of anti-matter erupting from lightning storms. Like AGILE, the spacecraft measured incredible energies, but its detectors also indicated that the gamma rays were colliding with positrons, which are their anti-matter counterparts.

[and]
Scientists detect dark lightning linked to visible lightning
Apr 24, 2013

http://phys.org/news/2013-04-scientists-dark-lightning-linked-visible.html

"Our results indicate that both these phenomena, dark and bright lightning, are intrinsic processes in the discharge of lightning," said Nikolai Østgaard, who is a space scientist at the University of Bergen in Norway and led the research team.

He and his collaborators describe their findings in an article recently accepted in Geophysical Research Letters—a journal of the American Geophysical Union.

Dark lightning is a burst of gamma rays produced during thunderstorms by extremely fast moving http://phys.org/tags/electrons/ colliding with http://phys.org/tags/air+molecules/ . Researchers refer to such a burst as a terrestrial gamma ray flash.

Dark lightning is the most energetic radiation produced naturally on Earth, but was unknown before 1991. While scientists now know that dark lightning naturally occurs in thunderstorms, they do not know how frequently these flashes take place or whether visible lightning always accompanies them.

Read more at: http://phys.org/news/2013-04-scientists-dark-lightning-linked-visible.html#jCp

 

[Ken G]

Dark lightning would appear to be an interesting phenomenon, but it probably does not involve quantum mechanics or the Pauli exclusion principle.

 

[anorlunda]

I've seen that question on here before a few times. It has come up relatively frequent. Interesting how everyone keeps coming to that conclusion! A good mathematical explanation is given at http://en.wikipedia.org/wiki/Identical_particles#Symmetrical_and_antisymmetrical_states.

Last year, I posted a nearly identical question., and I received the following reply

Anyway - it makes sense to start with http://en.wikipedia.org/wiki/Spin-statistics_theorem

Both of these Wikipedia articles are very helpful and responsive to the questions. Thank you HeavyMetal and tom.stoer.

But neither wiki article points to the other, and neither is linked in the "See Also" section of the PauliExclusion Principle wiki article. Also, neither title of the wiki articles suggests a relation to the question. Searching WIkipedia can be a nightmare.

My question is this: Can mentors offer laymen tips on searching Wikipedia to find answers to their own questions before posting to PF? Or can you point me to a PF thread discussing that question?

Apologies if this is off topic to the OP, but I hope it is a valid follow-up.