The Pauli Exclusion Principle: Exploring its Implications Across the Universe

[airydisc2004]

Hi

New to this forum. I am not a physicist (maths ), but have a healthy curiosity and interest in Quantum physics.

I have a question regarding the Pauli Exclusion Principle. From what I have understood previously, this applies to a single atom or maybe atoms in close proximity. So, no two fermions (e.g electrons) in a system can exist in the same quantum state. It is possible for two electrons to have the same quantum energy states providing they have opposite spin, but that's it.

In a book I am reading by Brian Cox and Jeff Foreshaw "Everything that can happen does Happen", it is very clear that they are saying that if an electron changes its energy state, somewhere, even the other side of the universe, another electron responds instantaneously so that the two electrons are not in the same energy state.

My questions are based on three levels.

1. Instantaneous response over distance violates Relativity, although they cover this in the book by saying that its only messaging at faster than c that is prohibited. Surely any response to an event or action is via messaging? Cause and effect? If we are saying that we have an electron in one place in the universe that changes its quantum state in response to another electron in another part of the universe changing its state, where does the trigger or cause for this change come from, in order to provide instantaneous change?

2. If we imagine that the entire universe can be considered as a single "system", then we can imagine a state where all energy levels are "known", (all quantum particles are in ownership of, or aware of the state or condition of all other particles). In which case, why is it that an electron can change its quantum state in the first place, because by doing so, it is moving into an energy state already occupied by another electron. If the change is instant, this should not happen should it?

3. Is there any merit in the idea that if there are, say, 10⁸¹ electrons in the universe, then there are 10⁸¹ quantum energy states available?

Please excuse the naivety, this is all new to me, but fascinating.

Airydisc

 

[Bill_K]

I have a question regarding the Pauli Exclusion Principle. From what I have understood previously, this applies to a single atom or maybe atoms in close proximity. So, no two fermions (e.g electrons) in a system can exist in the same quantum state.

More general than that, no two electrons anywhere can exist in the same quantum state. However 'state' includes more than just energy and spin.

In a book I am reading by Brian Cox and Jeff Foreshaw "Everything that can happen does Happen", it is very clear that they are saying that if an electron changes its energy state, somewhere, even the other side of the universe, another electron responds instantaneously so that the two electrons are not in the same energy state.

Sounds like something Brian Cox might say. But no, it's completely wrong. For example, every (un-ionized) Hydrogen atom in the universe has an electron in the ground state. They all have the same energy. But the reason that this is consistent with the Pauli principle is that each electron is localized, i.e. confined to the atom and its immediate neighborhood. That means they are all in different states, and except for perhaps its nearest neighbors the electron in one atom has no influence on any of the others.

 

[Jilang]

Yes, if they are in different places (on average) they do have different wavefunctions and are not in the same state.

 

[airydisc2004]

Hi. Thanks for the answer.

Does this mean that PEP does only really apply to single atoms and those in close proximity, so that in some respects its irrelevant what the energy state of one electron is compared to another over a large distance, because they can't communicate/message...or have I misunderstood your meaning?

airydisc2004

 

[Bill_K]

Does this mean that PEP does only really apply to single atoms and those in close proximity, so that in some respects its irrelevant what the energy state of one electron is compared to another over a large distance, because they can't communicate/message...or have I misunderstood your meaning?

Normally that's right. However see delocalized electrons, in which the wavefunction of a single electron can extend over several atoms.

 

[bhobba]

More general than that, no two electrons anywhere can exist in the same quantum state. However 'state' includes more than just energy and spin.

Yes - oh yes - very true.

The reason for my comment is since Brian Cox was mentioned he, in a lecture, once famously made some comments about heating up a diamond:

'So here’s the amazing thing: the exclusion principle still applies, so none of the electrons in the universe can sit in precisely the same energy level. But that must mean something very odd. See, let me take this diamond, and let me just heat it up a bit between my hands. Just gently warming it up, and put a bit of energy into it, so I’m shifting the electrons around. Some of the electrons are jumping into different energy levels. But this shift of the electron configuration inside the diamond has consequences, because the sum total of all the electrons in the universe must respect Pauli. Therefore, every electron around every atom in the universe must be shifted as I heat the diamond up to make sure that none of them end up in the same energy level. When I heat this diamond up all the electrons across the universe instantly but imperceptibly change their energy levels.'

I remember when I first heard it I thought - that can't be right - but for the life of me couldn't see the error. Then after a bit of thinking it hit me - the gap between any two real numbers allows for an infinite number of other real numbers - so what would happen - nothing - the new energy levels would simply fit in between the gaps.

And there is the issue that the state includes more than just energy which means position plays a role as well - an electron elsewhere can have the same energy - its really only a concern if their wave functions overlap and that doesn't happen for systems that are well separated.

Still the Pauli Exclusion Principle applies to any electrons anywhere and does lead to some puzzling thought provoking stuff that a bit of care is required to disentangle - and even professional physicists like Brian Cox can be 'fooled'.

Thanks
Bill

 

[airydisc2004]

Thanks for your answers.

Your answer about Brian Cox being fooled (from my naive position anyway) seems a little kind to him, given that it makes no sense to consider a scenario where there is instant messaging across vast distances. Surely this must have been an episode of "audience impressing", rather than simply having been fooled. Having said that, he does back it up with the same suggestions in the book I mentioned.
Can there be an infinite number of energy levels for an electron to exist in? If so, is my guess earlier about the number of electron energy levels in the universe equalling the number of electrons in the universe, or am I being a non-physicist :)

It is sooo difficult for non-physicists (non-mathematicians) like me to grasp Quantum theory properly, because the maths is vital, and I am getting the impression that its a question of "trust the maths and pay less attention to trying to visualise models". However, for those of us not having had a rigorous enough maths education (partly due to poor maths education standard, partly due to not paying attention), we have to rely on imaginary models and scenarios.

AD2004

 

[WannabeNewton]

Can there be an infinite number of energy levels for an electron to exist in?

It depends on the system being considered. For a hydrogen atom there are only finitely many discrete energy levels before the electron becomes free and "escapes" the atom. A free electron can take on a continuous range of energies in the reals. There can also be systems with discrete energy levels but infinitely many of them such as the infinite square well.

 

[bhobba]

Your answer about Brian Cox being fooled (from my naive position anyway) seems a little kind to him, given that it makes no sense to consider a scenario where there is instant messaging across vast distances. Surely this must have been an episode of "audience impressing", rather than simply having been fooled. Having said that, he does back it up with the same suggestions in the book I mentioned.

Actually IMHO fooled is a little unkind to him, which is why I put it in quotes.

From a simple model that Brian Cox referenced in his answer when the issue was raised it can be shown that it doesn't matter how far electrons are apart they can never have exactly the same energy level.

This is why I prefer my answer that the levels simply move to the always infinite spaces available, but its a far from trivial issue as Sean Carrols discussion shows:
http://www.preposterousuniverse.com/blog/2012/02/23/everything-is-connected/

QM is most certainly a non trivial subject (its real issue is not usually mentioned in the populist press - I will outline it at the end of the post), but these days its conceptual core is well understood:
http://www.scottaaronson.com/democritus/lec9.html

Once you understand its simply an extension of standard probability theory a lot of populist hand-wavy rubbish such as consciousness causes collapse and other 'misconceptions' disappear.

The real issue with QM is that its a generalized probability model about 'marks' (outcomes of observations etc etc) left here in an assumed common sense classical world. But that world is in fact quantum - so exactly how does a theory that at its very foundations assumes such a world explain it?

A lot of progress has been made in resolving that, but a few issues remain. If you are interested in pursuing that further, at your level a good book is Omnes - Understanding Quantum Mechanics:
https://www.amazon.com/dp/0691004358/?tag=pfamazon01-20

Thanks
Bill

 

[San K]

So what is the correct wording?

Is it:

No two electrons, in the universe, can exist in the same quantum state?
or
No two electrons, can have the same energy level

how is a quantum state defined?

How many quantum numbers are involved in specifying the quantum state? Is it four?

which of the 4 quantum numbers would be effected by change of atom/location?

 

[WannabeNewton]

how is a quantum state defined?

Mathematically, its just a normalized element of a given Hilbert space or a Rigged Hilbert space, as well as tensor products of different Hilbert spaces if we have multi-component systems such as the position degrees of freedom of two different non-interacting particles or a single particle with its spin and position degrees of freedom treated as two independet sub-systems-note both of these work only because the corresponding observables are compatible i.e. the associated operators commute.

How many quantum numbers are involved in specifying the quantum state? Is it four?

It depends on the system. In general you need as many quantum numbers as needed in order to prepare a unique state after successive measurements of finitely many different observables. More precisely, say we start with two observables $A$ and $B$ that are compatible i.e. $[\hat{A},\hat{B}] = 0$. We prepare a system (or an ensemble of such) in a state $|a,b \rangle$. Well it's entirely possible for the eigenvalues $a$ and $b$ to be degenerate so this state isn't unique. However there's a theorem which states we can keep adding more and more observables to our list until we have a set $\{A, B, C,...\}$ such that $|a,b,c,... \rangle$ is non-degenerate and with it we can prepare a unique state.

which of the 4 quantum numbers would be effected by change of atom/location?

Electrons in orbitals have rather intricate possibilities for orbital configurations so you might have to be more specific because it depends on the change in orbital configuration as well as the occupation number of the new orbital an electron is sent to.

 

[San K]

Thanks WannabeNewton.

Electrons in orbitals have rather intricate possibilities for orbital configurations so you might have to be more specific because it depends on the change in orbital configuration as well as the occupation number of the new orbital an electron is sent to.

I mean moving an electron from same orbital (level) in one atom to another atom (with same configuration, same occupation number etc).

For example: what is the difference between -
an electron A in atom A of hydrogen
than
an electron B in the same orbital of atom B of hydrogen.

how/why is their quantum state different? which quantum number has changed?

I hope my question makes sense or let me know if I am missing some pieces of knowledge.

 

[bhobba]

For example: what is the difference between -
an electron A in atom A of hydrogen than an electron B in the same orbital of atom B of hydrogen. how/why is their quantum state different? which quantum number has changed?

The correct statement of the Pauli exclusion principle is 'the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles.'
http://en.wikipedia.org/wiki/Pauli_exclusion_principle

This only has consequences if they are in the same state.

To see why check out:
http://www.physics.ohio-state.edu/~eric/teaching_files/writing.course/sample3.shortdraft.pdf

The issue for two electrons by themselves it leads to a wavefunction that is zero if they are in the same state.

This applies to any electrons - anywhere in the universe.

Now for some caveats. That same state also means the state of the other particles in systems they are bound to such as atoms must be taken into account. This allows the vast majority of electrons to be in blissful ignorance of all the other electrons - except those in the same atom etc. Then we have entanglement - most other electrons are entangled with all sorts of other things - that makes it even harder for them to be in the same quantum state.

Sean Carroll examined the whole issue here:
http://www.physics.ohio-state.edu/~eric/teaching_files/writing.course/sample3.shortdraft.pdf

Personally though, like I said, to me its a non issue - since all you need to do is assume the energy levels slot into the gaps that are always there without even worrying if the exclusion principle has consequences or not.

Thanks
Bill