The Born interpration and electrostatic repulsion

Labarbapostiza

Just to get a discussion going on something that confuses me. (a few things)


If the Born interpretation is correct; that the Pauli exclusion principle is a case of the Born interpretation. If electrostatic repulsion is a result of Pauli's exclusion principle, then is it a result of Born's interpretation?

Morbert

Labarbapostiza wrote: »

Just to get a discussion going on something that confuses me. (a few things)


If the Born interpretation is correct; that the Pauli exclusion principle is a case of the Born interpretation. If electrostatic repulsion is a result of Pauli's exclusion principle, then is it a result of Born's interpretation?

The Pauli exclusion principle is not a case of the Born interpretation. The pauli exclusion principle is a result of fermion correlation.

Electrostatic repulsion is a result of the electromagentic force, which is unrelated to the pauli exclusion principle.

Labarbapostiza

Morbert wrote: »

The Pauli exclusion principle is not a case of the Born interpretation. The pauli exclusion principle is a result of fermion correlation..

Remember I am confused by all this. I believe Pauli's principle was not initially a quantum theory - but that did follow.

Looking at the Wikipedia page for the exclusion principle. For fermions, the superposition of two fermions gives a probability of zero.

What exactly does this mean? That if you tried to superimpose two fermions in the same state, something would force them apart?

Electrostatic repulsion is a result of the electromagentic force, which is unrelated to the pauli exclusion principle.

Okay, I'll explain my confusion or what I'm trying to think about. If the Born interpretation, interprets a wave function to be a probability distribution, then (and this is where I could be hugely wrong) the electric field density of an electron, is correlated to its' probability distribution. Do you see what I'm getting at?

Morbert

Labarbapostiza wrote: »

Remember I am confused by all this. I believe Pauli's principle was not initially a quantum theory - but that did follow.

Looking at the Wikipedia page for the exclusion principle. For fermions, the superposition of two fermions gives a probability of zero.

What exactly does this mean? That if you tried to superimpose two fermions in the same state, something would force them apart?

The correlation responsible for the Pauli exclusion principle does not have a classical analogue, so it cannot be understood in terms of a simpler classical concept like "force". It is simply a result of the fundamental and qualitatively different nature of quantum physics.

In quantum mechanics, a system is described with a state vector. This vector codifies all probabilities associated with the observables of the system. One of the features of these state vectors is that Fermions have 0 probability of being found in the same quantum state, and have diminishing probability of being found in similar states. This isn't a force, it's a statistical correlation inherent in quantum mechanics.

Okay, I'll explain my confusion or what I'm trying to think about. If the Born interpretation, interprets a wave function to be a probability distribution, then (and this is where I could be hugely wrong) the electric field density of an electron, is correlated to its' probability distribution. Do you see what I'm getting at?

The charge density of an electron is indeed the wavefunction (squared). Specifically, if you expand the state vector in a position basis, you will get a function over space that gives you the charge distribution.

What this means in the context of the Born interpretation is, if you set up many identical experiments where you probe the system for an electron, you will most frequently find an electron where the squared value of the wavefunction is greatest. Hence, the charge density of the system will be highest at these regions, because that is where the electrons (and hence the charge) most frequently is.

As an aside, the Born interpretation is a very good interpretation of quantum mechanics. The only interpretation that I find more compelling is the consistent histories interpretation, and there is not a huge amount of difference between the two.

Labarbapostiza

Morbert wrote: »

The correlation responsible for the Pauli exclusion principle does not have a classical analogue, so it cannot be understood in terms of a simpler classical concept like "force". It is simply a result of the fundamental and qualitatively different nature of quantum physics.

In this instance I'm interested in the quantum interpretation of the classical phenomenon.

In quantum mechanics, a system is described with a state vector. This
vector codifies all probabilities associated with the observables of the system. One of the features of these state vectors is that Fermions have 0 probability of being found in the same quantum state, and have diminishing probability of being found in similar states. This isn't a force, it's a statistical correlation inherent in quantum mechanics.

I understood that the state vectors for Fermions had a zero probability of being found in the same quantum state. But I'm sure what it means. If the two fermions are superimposed, do they physically vanish. Or does it mean that since the probability is zero of them being in the same state, they <i>have to be</i> somewhere else?

I've read some literature which I don't have to hand, where it's explained that as the electrons come close to super-imposing, the uncertainty principle means they never quite fully super-impose. I am uncertain. Also that the become indistinguishable; that as you push them closer together, instead of overlapping, they skip that stage and switch places.

The charge density of an electron is indeed the wavefunction (squared). Specifically, if you expand the state vector in a position basis, you will get a function over space that gives you the charge distribution.

What this means in the context of the Born interpretation is, if you set up many identical experiments where you probe the system for an electron, you will most frequently find an electron where the squared value of the wavefunction is greatest. Hence, the charge density of the system will be highest at these regions, because that is where the electrons (and hence the charge) most frequently is.

Okay, I'm just going to describe an experiment, and you can tell me where it's wrong or how I'm wrong in my interpretation.

Take two metal spheres and charge them; place an excess of delocalised electrons on them. Move them together, and there will be a measurable force between the two - as given by the classical formula.

If the wave function squared gives the charge density of an electron; an the observance would be that the more electrons the greater the charge density of the - the electrons are not simply on the surface of the sphere, but there is a probability they are spread out across space (that probability diminishing with distance) - Could you confirm with me if that is a correct assumption.

My next assumption, is there are enough electrons to give the impression of a field; essentially the classical electric field. (and quantum tunnelling being in effect the probability that some of the electrons being in space, a distance from a charged surface).

Now, this is the bit were it gets incredibly shaky. By moving the charged spheres together, I am attempting to superimpose fermions - there is a zero probability of them being superimpose; Where are they? Is the force experienced the Born interpretation refusing to be broken, and giving back the force I have added in repulsion.

I know there is something wrong about this. I'm trying to reconcile several different ideas. I know the force carriers for electrons are photons. My patchy understanding before, was the charge density of an electron (or even a charged sphere) was a large density of virtual photons cancelling each other. By adding energy to the system (bringing the two charged spheres together) Some of the photons, equivalent in energy to the energy added to the system, repel.

My understanding of electric current in a conductor, and electric charge on a surface (the potential difference;the voltage; between that surface and the ground), being a case of the Pauli exclusion principle.

Obviously, I am very wrong somewhere - or a few places even.

As an aside, the Born interpretation is a very good interpretation of quantum mechanics. The only interpretation that I find more compelling is the consistent histories interpretation, and there is not a huge amount of
difference between the two.

I think there's some underlying magic to Max Born's interpretation. That probability itself <i>is</i> the structure and substance of matter and energy.

Morbert

Labarbapostiza wrote: »

In this instance I'm interested in the quantum interpretation of the classical phenomenon.

But there is no classical phenomenon of exclusion. The Pauli exclusion principle does not exist in classical mechanics.

I understood that the state vectors for Fermions had a zero probability of being found in the same quantum state. But I'm sure what it means. If the two fermions are superimposed, do they physically vanish. Or does it mean that since the probability is zero of them being in the same state, they <i>have to be</i> somewhere else?

The latter. You cannot have Fermions occupying the same state. They have to be in different states. This does not mean they have to be somewhere else, however. For example, two electrons with different spin states do not have to avoid each other to obey the exclusion principle, because even if they are in the same place, they have different spin and therefore different quantum states (They will, however, attempt to avoid each other because of the unrelated Coulomb force.)

I've read some literature which I don't have to hand, where it's explained that as the electrons come close to super-imposing, the uncertainty principle means they never quite fully super-impose. I am uncertain. Also that the become indistinguishable; that as you push them closer together, instead of overlapping, they skip that stage and switch places.

What indistinguishable means in the context of quantum mechanics is the observables of a system do not change if any 2 particles swap or "exchange" position. Analogously, all the letter "a"s in this paragraph are indistinguishable. If I swap two around, the paragraph remains the same. Electrons are always indistinguishable, not just when they are nearby.

Okay, I'm just going to describe an experiment, and you can tell me where it's wrong or how I'm wrong in my interpretation.

Take two metal spheres and charge them; place an excess of delocalised electrons on them. Move them together, and there will be a measurable force between the two - as given by the classical formula.

If the wave function squared gives the charge density of an electron; an the observance would be that the more electrons the greater the charge density of the - the electrons are not simply on the surface of the sphere, but there is a probability they are spread out across space (that probability diminishing with distance) - Could you confirm with me if that is a correct assumption.

My next assumption, is there are enough electrons to give the impression of a field; essentially the classical electric field.

The bit in bold is not correct. The electric field is not made of electrons. That would be like saying the gravitational field is made of skydivers. Instead, the electric field is a component of the electromagnetic field. The particle associated with the electromagnetic field is the photon. The field associated with electrons is called the dirac field. But it is not important in your particular thought experiment.

Now, this is the bit were it gets incredibly shaky. By moving the charged spheres together, I am attempting to superimpose fermions - there is a zero probability of them being superimpose; Where are they? Is the force experienced the Born interpretation refusing to be broken, and giving back the force I have added in repulsion.

In this case, the force experienced would be the coulomb force. This is not merely a correlation, it is an actual force and does have a classical analogue. I.e. Even if the exclusion principle was suspended for this instance, there would still be a force, and the spheres would still resist coming close together.

I know there is something wrong about this. I'm trying to reconcile several different ideas. I know the force carriers for electrons are photons. My patchy understanding before, was the charge density of an electron (or even a charged sphere) was a large density of virtual photons cancelling each other. By adding energy to the system (bringing the two charged spheres together) Some of the photons, equivalent in energy to the energy added to the system, repel.

Charge density simply means density of electrons. It is simply a property of the electron, a susceptibility. The photons don't carry charge. They instead mediate interactions between charged particles.

My understanding of electric current in a conductor, and electric charge on a surface (the potential difference;the voltage; between that surface and the ground), being a case of the Pauli exclusion principle.

Obviously, I am very wrong somewhere - or a few places even.

The repulsion can be divided into two components: Repulsion due to a force (the exchange of virtual photons between electrons), and "repulsion" due to an inherent correlation between electrons, which is not due to any force. In your thought experiment, the former dominates. A case where the latter dominates is in the stiffness of material in general. The atomic structure and properties of material, and the fact that you don't fall through your chair, are all largely due to the exclusion principle.

I think there's some underlying magic to Max Born's interpretation. That probability itself <i>is</i> the structure and substance of matter and energy.

I wouldn't agree with that description. The Born interpretation does not say reality is probability itself. Rather, it says the statements we make about reality must be probabilistic, and must pertain to ensembles. I.e. We cannot say an electron is at position x. We can only say, if we carry out an experiment multiple times, the probability that an electron is at position x is p(x). Probability is still a tool to describe reality, and not reality itself.

Another way of saying this is as follows: According to the Born interpretation, the wavefunction is not physically real. It is merely a mathematical tool that allows us to calculate the frequency of different experimental outcomes when the same experiment is carried out multiple times.

Labarbapostiza

Morbert wrote: »

But there is no classical phenomenon of exclusion. The Pauli exclusion principle does not exist in classical mechanics.

I didn't mean classical mechanics. I meant Pauli's principle was not initially quantum mechanical. Classical chemistry is as different to modern chemistry as classical physics is to modern physics. I believe before Pauli, they were teaching in chemistry that the nucleus had hooks, which electrons attached to.

The latter. You cannot have Fermions occupying the same state. They have
to be in different states. This does not mean they have to be somewhere else,
however. For example, two electrons with different spin states do not have to
avoid each other to obey the exclusion principle, because even if they are in
the same place, they have different spin and therefore different quantum states
(They will, however, attempt to avoid each other because of the unrelated
Coulomb force.)

Yes, I was aware two electrons could be in the same physical place if they had opposite spin.

But, what I'm trying to understand is, what is the Coulomb force. And I mean that in, what is the QM explanation of the Coulomb force? This is in fact my principle question.

What indistinguishable means in the context of quantum mechanics is the
observables of a system do not change if any 2 particles swap or "exchange"
position. Analogously, all the letter "a"s in this paragraph are indistinguishable. If I swap two around, the paragraph remains the same.
Electrons are always indistinguishable, not just when they are nearby.

I know electrons are indistinguishable from each other, as in all electrons are electrons. But, what I meant, and I base this on looking at a diagram in a book I do not have, as you overlap the probability densities of two electrons, each electron has a probability of being in the other's density (wave function), then they're indistinguishable in the sense that one electron could be in the other's wave function, and if that is so then the other must be in the other's wave function. I could have this completely wrong - I didn't have the book for long.

The bit in bold is not correct. The electric field is not made of electrons.
That would be like saying the gravitational field is made of skydivers.
Instead, the electric field is a component of the electromagnetic field. The
particle associated with the electromagnetic field is the photon. The field
associated with electrons is called the dirac field. But it is not important in
your particular thought experiment.

I didn't really mean it in the sense of the electric field, as the modern interpretation. More the observed invisible field around a charge object before modern theory.


The electron in is a Dirac field or Fermionic field; which means two electrons can not be in the same place unless they have opposite spin. What I'm trying to understand is how this field joins up with the electromagnetic field.

Morbert

What (I think) is relevant to your question is the distinction between the Pauli Exclusion principle and electrostatic repulsion.

The Pauli exclusion principle is a statistical phenomenon, a result of the inherent qualitative difference between quantum and classical correlation.

The electrostatic repulsion is a force, mediated through the exchange of virtal photons as they are emitted and absorbed by electrons. It is explicitly described by a two-body operator in the Hamiltonian. Quantum electrodynamics is the theory that describes the coupling of electrons to the electromagnetic field.

If you somehow eliminated the coupling, electrostatic repulsion would disappear. The electron correlation arisisng from the Exclusion principle would not.

Basic quantum chemistry calculations are good at capturing the former (correlation arising from the Exclusion principle). Modelling the latter is harder, and is still an active area of research.

Labarbapostiza

Morbert wrote: »

The electrostatic repulsion is a force, mediated through the exchange of virtal photons as they are emitted and absorbed by electrons. It is explicitly described by a two-body operator in the Hamiltonian. Quantum electrodynamics is the theory that describes the coupling of electrons to the electromagnetic field.

Sorry for taking so long to reply. I'd like to keep the discussion going, as I have a few more questions to ask.

If electrostatic repulsion is the coupling of electrons, via photons. How does the coupling work with electrostatic attraction; the electron to the proton?

The other thing you could explain is how the photon acts as its' own anti-particle.

Again, sorry for the delay.

Morbert

Labarbapostiza wrote: »

If electrostatic repulsion is the coupling of electrons, via photons. How does the coupling work with electrostatic attraction; the electron to the proton?

The coupling is between the electron and the electromagnetic field. I.e. Two electrons both couple to the electromagnetic field, and therefore induce "ripples" or excitation in the electromagnetic field as they move. These ripples propagate and act to push the electrons apart. Similarly, an electron and a proton both produce ripples, only this time the ripples act to pull the particles together.

Why the ripples push electron-electron pairs apart and pull electron-proton pairs together is a matter of how the charged particles (or, in field theory terms, the charged particle fields) couple to the electromagnetic field. It more or less amounts to a change in sign when you draw the Feynman diagrams.

The other thing you could explain is how the photon acts as its' own anti-particle.

This is also a consequence of quantum electrodynamics. When you draw Feynman diagrams and apply the rules accordingly, the photon will have identical properties whether it is moving forward in time or backwards in time. So it is its own anti-particle.

Labarbapostiza

Morbert wrote: »

The coupling is between the electron and the electromagnetic field. I.e. Two electrons both couple to the electromagnetic field, and therefore induce "ripples" or excitation in the electromagnetic field as they move. These ripples propagate and act to push the electrons apart. Similarly, an electron and a proton both produce ripples, only this time the ripples act to pull the particles together..

I don't have a thorough understanding of QED, but I know it uses the fine structure constant as the coupling constant to couple the electron field to the electromagnetic. Just jumping back to the Born Interpretation for a moment: if the amplitude of the wave function representing an electron are the probabilities of finding the electron in that position, .......this is the bit that really confuses me; Colomb's law...I know it's classical but, but by Colomb's law, the magnitude of the force of electrostatic repulsion between charged particles is inversely proportional to the square of the distance between them. The energies of the photons exchanged will be different, depending on the positions of the electrons.......I'm not sure if I really have a bad understanding of this.

Why the ripples push electron-electron pairs apart and pull electron-proton
pairs together is a matter of how the charged particles (or, in field theory
terms, the charged particle fields) couple to the electromagnetic field. It
more or less amounts to a change in sign when you draw the Feynman diagrams.

I understand that inverting the sign of the charge can give attraction. But, I can understand the photon having a repulsive effect.. I don't understand how it can attract. I have my own little theory, but what is the generally accepted theory that explains how the photons in the case of a proton and electron attract.

And another point. I'll use the example of an electron orbiting in a proton. The electron can jump energy levels through being energised (absorbing photons)......Until it can completely escape the proton. Does Colomb's law apply here. That the higher the energy of the electron means it's physically located further away from the proton. I have a mish mash of various explanations in my head, and they don't all add up, or more they add to the confusion.

This is also a consequence of quantum electrodynamics. When you draw Feynman diagrams and apply the rules accordingly, the photon will have identical properties whether it is moving forward in time or backwards in time. So it is its own anti-particle.

But using the same definition, can you say the electron is its' own antiparticle?

Morbert

Labarbapostiza wrote: »

I don't have a thorough understanding of QED, but I know it uses the fine structure constant as the coupling constant to couple the electron field to the electromagnetic. Just jumping back to the Born Interpretation for a moment: if the amplitude of the wave function representing an electron are the probabilities of finding the electron in that position, .......this is the bit that really confuses me; Colomb's law...I know it's classical but, but by Colomb's law, the magnitude of the force of electrostatic repulsion between charged particles is inversely proportional to the square of the distance between them. The energies of the photons exchanged will be different, depending on the positions of the electrons.......I'm not sure if I really have a bad understanding of this.

A derivation of the inverse squared law from the exchange of virtual photons also requires quantum electrodynamics (it always boils down to electrodynamics). You would need to calculate the the probability amplitude for the exchange of a virtual photon between two point charges, then calculate the energy of the interaction, and observe the dependence of the interaction on the distance between the particles.

There isn't really any intuitive understanding of exchange particles beyond the rules of quantum electrodynamics. I would recommend this book if you are interested in picking up the fundamentals.

I understand that inverting the sign of the charge can give attraction. But, I can understand the photon having a repulsive effect.. I don't understand how it can attract. I have my own little theory, but what is the generally accepted theory that explains how the photons in the case of a proton and electron attract.

And another point. I'll use the example of an electron orbiting in a proton. The electron can jump energy levels through being energised (absorbing photons)......Until it can completely escape the proton. Does Colomb's law apply here. That the higher the energy of the electron means it's physically located further away from the proton. I have a mish mash of various explanations in my head, and they don't all add up, or more they add to the confusion.

The generally accepted theory is quantum electrodynamics.

It would be a mistake think of the exchange of photons as some classical mechanical process, like the throwing of a medicine ball between two people. If a similar classical scenario insisted upon, you could imagine two people facing away from each other and tossing a boomerang between each other. Each time a person catches or throws the boomerang, they are pushed towards the other person. Or, perhaps more appropriately, you could imagine the tossing of the medicine ball with time reversed.

But I must stress that this scenario has little to do with the exchange of virtual photons. The deepest understanding of the process can only come from an understanding of the rules that govern the emission, absorption, and propagation of virtual photons.

But using the same definition, can you say the electron is its' own antiparticle?

If you apply the same rules to an electron, you observe that the properties of the electron change when it is propagated backward through time. This permits the demarcation of the particles into electrons, with minus charge, and positrons, with plus charge.

Labarbapostiza

Morbert wrote: »

What (I think) is relevant to your question is the distinction between the Pauli Exclusion principle and electrostatic repulsion.

The Pauli exclusion principle is a statistical phenomenon, a result of the inherent qualitative difference between quantum and classical correlation.

If you somehow eliminated the coupling, electrostatic repulsion would disappear. The electron correlation arisisng from the Exclusion principle would not.

To jump back to one of your earlier comments. I want to respond to some of your other points, but I'm lacking the mental energy to do it at the moment (mental confusion demands a lot of energy).

But........Philip Moriarty on the latest Sixty Symbols video "Do Atoms ever Touch".

At about 2 minutes, 52 seconds, he says "And those discussion on the internet are quite correct. There is a repulsive force, and it is ultimate due to something called the Pauli Exclusion principle"

https://www.youtube.com/watch?v=P0TNJrTlbBQ

To be quite honest, I'm confused at this moment in time.

Labarbapostiza

Morbert wrote: »

It would be a mistake think of the exchange of photons as some classical mechanical process, like the throwing of a medicine ball between two people. If a similar classical scenario insisted upon, you could imagine two people facing away from each other and tossing a boomerang between each other. Each time a person catches or throws the boomerang, they are pushed towards the other person. Or, perhaps more appropriately, you could imagine the tossing of the medicine ball with time reversed.

Yes, I've heard those descriptions before, I don't think they come anywhere near describing what is happening.

Reversing the time on your photons works as an explanation in so far as you can now say the photons appear to attract when you run the camera backwards. I can see it working on paper; I have seen the Feynman diagrams. Now. In a quantum spacial fluctuation, in both halves of the space, time will be experienced as going in one direction, we'll say forward, but relative to each other, time will be reversed. So, in the life span of the fluctuation, two photons (at the minimum) will be created in the electromagnetic field. For both, they will experience time running forward, as the space expands, and then experience time running backwards as the space contracts; but relative to each other their time will always be reversed.

If our universe is the product of a quantum fluctuation, then their could be a mirror image with time running in reverse relative to our half of the space. Reversing the camera on photons to make them pull instead of push, it will give you the right answer, but is the explanation wrong. Why I say this is our special dimensions are observably isotropic; we can move backwards and forwards through them, but our time dimension is not; it is if you reverse the camera (to a centre extent; entropy appears to move in one direction), time appears to be isotropic, but it's an artificial isotropy.

Morbert

Labarbapostiza wrote: »

Yes, I've heard those descriptions before, I don't think they come anywhere near describing what is happening.

Yes, that is precisely my point. It is quantum electrodynamics, not classical mechanics, that describes what is happening. It is impossible to understand these interactions using classical concepts. If you think of a photon being "passed" between electrons, you will end up conjuring the notion of the photon exerting a "force" on the electrons, but force is the very thing we are trying to explain. So instead, we say the photons are gauge bosons that interact with electrons and protons (as described by QED) in such a way that the conservation of momentum results in attraction and repulsion.

You also don't need to invoke spacetime fluctuation to understand the interactions. You just need quantum electrodynamics. It might be unsatisfying to reduce a phenomenon to a set of mathematical rules, but those rules are ultimately the deepest understanding we will have.

At about 2 minutes, 52 seconds, he says "And those discussion on the internet are quite correct. There is a repulsive force, and it is ultimate due to something called the Pauli Exclusion principle"

Yes, the pauli-exclusion principle plays a major role in the impenetrability of materials. The use of the word force is ok, provided it is understood that the "force" of the exclusion principle is not due to the exchange of a photon or other gauge boson. Instead, it is due to the fermi-dirac statistics of electrons and their indistinguishability. So even if the electromagnetic field ceased to exist, the pauli "force" discussed in the video would still exist. I.e. The electrons do not need to exchange photons to "repel" or "exclude" each other. Also, what little hope we had of drawing a classical analogy for the electromagnetic force disappears completely when trying to conceptualise the exclusion principle, which is a wholly quantum phenomenon.

The terminology can actually be quite confusing. The "force" due to the pauli exclusion principle is an example of an "exchange interaction", while the force due to the exchange of photons between electrons (and protons) is an example of an "exchange force". Both involve exchange, and both can be interpreted as a force, but they have very different mechanisms, and "force" is often reserved for those processes involving the exchange of gauge bosons.

[edit] - When I get the chance, I will go into more detail, and explain why fermi-dirac statistics and indistinguishability gives rise to the pauli exclusion principle. I have a feeling if I explained exchange forces and exchange interactions in terms of Feynmann diagrams, the distinction would be clearer.

Labarbapostiza

Morbert wrote: »

The terminology can actually be quite confusing. The "force" due to the pauli exclusion principle is an example of an "exchange interaction", while the force due to the exchange of photons between electrons (and protons) is an example of an "exchange force". Both involve exchange, and both can be interpreted as a force, but they have very different mechanisms, and "force" is often reserved for those processes involving the exchange of gauge bosons.

But as I understand, both bosons and fermions have exchange interactions. And the difference that distinguishes each class, and leads to the Pauli exclusion for fermions, is that bosons have symmetric wave functions, and fermions have anti-symmetric wave functions.

From memory (I might get this wrong as it's from glancing in text book)

Take two wave functions [latex]\Psi[/latex], we'll call 1 and 2. Superimposing them is the product of 1 and 2.

For bosons, with symmetric WFs

[latex]\Psi(1,2) = \Psi(2,1)[/latex]

For fermions with anti-symmetric wave functions

[latex]\Psi(1,2) = -\Psi(2,1)[/latex]

I don't have the luxury of stack of good text books to explore this in detail. I think I've seen this graphed, in another book I do not have. But that the anti-symmetry performs as a barrier for fermions; but I'm not sure if particle exchange plays any part in the barrier, only that the distance two fermions can exchange has to be much wider than bosons, which can completely overlap.

It's also very interesting that since Helium is symmetric, it has boson like qualities. All the weird stuff it can do when its' super cooled.

Is the whole trick of super conductivity getting Fermions to form symmetric waves; so they can form Bose-Einstein condensates?

I'll have to learn how to shoplift, or get access to a good library.

Morbert

Labarbapostiza wrote: »

But as I understand, both bosons and fermions have exchange interactions. And the difference that distinguishes each class, and leads to the Pauli exclusion for fermions, is that bosons have symmetric wave functions, and fermions have anti-symmetric wave functions.

From memory (I might get this wrong as it's from glancing in text book)

Take two wave functions [latex]\Psi[/latex], we'll call 1 and 2. Superimposing them is the product of 1 and 2.

For bosons, with symmetric WFs

[latex]\Psi(1,2) = \Psi(2,1)[/latex]

For fermions with anti-symmetric wave functions

[latex]\Psi(1,2) = -\Psi(2,1)[/latex]

A much better term than exchange interaction is "exchange correlation". Let's write out the fermion wavefunction more explicitly.

[latex]\Psi(1,2) = \psi_1(1)\psi_2(2) - \psi_1(2)\psi_2(1)[/latex]

where [latex]\psi_1[/latex] and [latex]\psi_2[/latex] are the states of the individual particles.

This wavefunction is antisymmetric

[latex]\Psi(2,1) = \psi_1(2)\psi_2(1) - \psi_1(1)\psi_2(2)[/latex]

[latex] = -(\psi_1(1)\psi_2(2) - \psi_1(2)\psi_2(1))[/latex]

[latex]= -\Psi(1,2)[/latex]


The Pauli exclusion principle says "no two fermions can be in the same quantum state". So let's see what happens if we try. Let's say [latex]\psi_1 = \psi_2[/latex], i.e. [latex]\psi_1[/latex] is the same state as [latex]\psi_2[/latex]. Then we have

[latex]\Psi(1,2) = \psi_1(1)\psi_2(2) - \psi_1(2)\psi_2(1)[/latex]

[latex]= \psi_1(1)\psi_1(2) - \psi_1(2)\psi_1(1)[/latex]

[latex]= \psi_1(1)\psi_1(2) - \psi_1(1)\psi_1(2)[/latex]

[latex]= \psi_1(1)\psi_1(2)(1-1)[/latex]

[latex]= 0[/latex]

So the probability of finding two fermions in the same quantum state is 0. Notice that this is simply a statistical feature of the wavefunction. It is inherent in the quantum description of the system. The electromagnetic force, on the other hand, really is a dynamical, physical process whereby virtual photons travel between the two electrons (or an electron and proton).

but I'm not sure if particle exchange plays any part

It doesn't. The exclusion principle does not involve any particle exchange. I should have been clearer in my original post, where I said both involved exchange.The confusion is in the word "exchange". Below are the two contexts.

The electromagentic force between two electrons is due to the exchange of virtual photons (I.e. virtual photons travel back and forth between the electrons).

The Pauli exclusion principle is due to the fact that the electrons are correlated and indistinguishable. A feature of this correlation is the wavefunction changes sign if the two electrons are swapped or exchanged.

In the former, the exchange of a particle refers to the emission of a particle by one electron, and the absorption of the same particle by the other. In the latter, exchange refers to a symmetry quality of the wavefunction when two particles are swapped.

Labarbapostiza

Morbert wrote: »

So the probability of finding two fermions in the same quantum state is 0.

I've been trying to think of a good question. It's not that I can't, it's more too many questions.

In Beiser's Concepts of modern physics, in the chapter on the Pauli Exclusion principle, it says, that Helium atoms are symmetric.

Does this make them bosons?..........Even though they are composed of fermions. Can they pass right through each other?

The behaviour of super-cooled helium condensate. It can appear to flow against the force of gravity.....is it? When the helium passes through glass, etc, is this because it is symmetric?

What does anti-symmetric mean in terms of a fermion's wavefunction ? What would the distinction between being asymmetric as opposed to anti-symmetric mean?