Understanding Spin: A Basic Explanation for Beginners in Physics

[Nugatory]

I liked Ken's suggestion of a mysterious property of an electron.
Can you please explain to my simplistic way of thinking why the Pauli principle fails in this case.

It does not "fail". The Pauli principle says that two fermions may not have identical quantum numbers, so it says that you can have two electrons in different orbitals, or two electrons in the same orbital with different spins, but not two electrons in the same orbital with the same spins.

 

[Nugatory]

I think that I can visualise the classical approach to the recombination of hydrogen atoms; only three body collisions between hydrogen atoms and a third body to absorb the heat of combination are successful in forming the molecule when the the electron spins are in opposition. When the spins are similar the 3 entities fly apart.

That's not right. There's no third body required or involved, the energy released by the reaction is not heat (although it may end up as heat eventually, and generally will if our reacting atoms are surrounded by matter). Most important, no matter what the initial state of the electrons we end up with the lowest orbital filled and the electrons having opposite spins - there is no "flying apart" if the spin are in opposition, they just change to allow the interaction to proceed.

Consider now the case of an electron attempting via Coulomb attraction to join an unfilled orbit of an ion. What prevents it from violating the Pauli principle?

You are trying to visualize this with classical thinking, imagining that the electron is a little object that moves around and can approach an ion, or that we could in principle try to force an electron with the wrong spin into an orbital.

However, from a quantum mechanical point of view, the two electrons are not separate particles. They are a single quantum system with a single quantum state, and none of the possible states have two electrons in the same orbital with the same spin. It's like the top and bottom of a rolling wheel; the entire wheel is moving and no matter how it moves the top and bottom will be opposite. It makes no sense to ask if we could make the top of the wheel be at one side while leaving the bottom at the bottom.

 

[nettleton]

'Staff mentor' I am still puzzled by the idea ' that they just change to allow the interaction to proceed'. If that is the action what is the cause?

 

[Nugatory]

'Staff mentor' I am still puzzled by the idea ' that they just change to allow the interaction to proceed'. If that is the action what is the cause?

You are still trying to think of the two electrons as two separate things each with their own spin, so that something has to "cause" one of them to change its spin before the two of them can settle into the same orbital.

I know of no good classical analogies for multi-particle quantum systems (even the phrase "multi-particle" doesn't mean quite what it sounds like, because the word "particle" doesn't mean what it does in ordinary language). The closest I've been able to come up with is the example of a rotating wheel: If you think of two opposite sides of the wheel as two independent objects, it is very mysterious that somehow they are always moving in opposite directions. Once you realize that there is really just one thing there, the wheel, it makes more sense; if the two sides didn't always move in opposite directions it wouldn't be a wheel.

 

[vanhees71]

In other words a many-electron system is always entangled due to the fact that they are indistinguishable fermions and the state must be antisymmetric under interchanging any pair of the electrons (which defines what a fermion is). So it doesn't make sense to talk about any individual electron but you have to consider the entire system of many electrons as a whole.

 

[Khashishi]

It might be useful to look at orthohelium and parahelium. Orthohelium is helium in a state where both electrons have the same spin. Parahelium is helium in a state where electrons have opposite spin. (Normal ground state helium is parahelium.)
There are useful Grotrian diagrams here: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/helium.html

If electrons with the same spin were repelling each other with a "force", we'd expect the orthohelium states to have higher energies. Actually they have lower energies. There simply is no state of orthohelium where both electrons are in the 1s state. It's just not a state.

 

[nettleton]

I am beginning to think that I am entering a Zeno paradox. Thus, at what distances does the system of wave functions begin their interaction and how close is this to entanglement?

 

[Khashishi]

Wavefunction, not wavefunctions. There is one wavefunction to describe both electrons.

 

[houlahound]

I believe the frustration emerges from the pioneers choosing the only language they had available to hook new concepts to.

Spin implies something spinning.

Now we know more a new language could be adopted to better reflect our understanding. It would be very productive IMO.

 

[Nugatory]

I believe the frustration emerges from the pioneers choosing the only language they had available to hook new concepts to.

Yes, and we have similar problems with the words "particle" and "observation" - both came into general use before the non-classical nature of the phenomena they attached to was fully understood.

It's a source of never-ending misery for the explainers as well as the explainees

 

[houlahound]

Make a good Insights article, a lexicon of nonclassical objects and phenomenon.

 

[nettleton]

Thanks for all the help offered. However, I am still puzzled by the distance over which two electrons can share a single wave function.

 

[Khashishi]

Basically, if the wavefunctions of the individual electrons don't overlap at all (or negligibly), you can treat them as separate wavefunctions. But if they interact at all, then you have to redo the calculation with a single wavefunction.

 

[akvadrako]

Yes, and we have similar problems with the words "particle" and "observation" - both came into general use before the non-classical nature of the phenomena they attached to was fully understood.

It's a source of never-ending misery for the explainers as well as the explainees

Then why not start coming up with a better language? Seems like you are in the perfect position to ignite such an endeavour. I've been lurking here for a while and it does get annoying to see the same endless arguments over unclear language and the misunderstanding it introduces. Just a suggestion, maybe a kind of moderated wiki of definitions would help.

 

[houlahound]

It's not a simple matter as writing a glossary. It more like writing a functional programming language from scratch. Albeit a most unnatural language to boot.

 

[vanhees71]

Then why not start coming up with a better language? Seems like you are in the perfect position to ignite such an endeavour. I've been lurking here for a while and it does get annoying to see the same endless arguments over unclear language and the misunderstanding it introduces. Just a suggestion, maybe a kind of moderated wiki of definitions would help.

The only good language is math, and the wave function of a many-body system (btw only very special cases of many-body systems can be described with a single wave function, namely only such, where the particle number is conserved; usually you need quantum field theory) is a function $\psi(t,\vec{x}_1,\vec{x}_2,\ldots,\vec{x}_N)$, where $N$ is the number of particles. It's meaning is that $|\psi|^2$ is the probability to find simultanaeously a particle at $\vec{x}_1$, $\vec{x}_2$, ..., $\vec{x}_N$. If the particles are indistinguishable, i.e., all their intrinsic quantum numbers (mass, spin, electric charges, lepton and baryon number) are the same, the wave function must also be unchanged or flip its sign whenever two of the particles are interchanged, i.e., $\psi(t,\vec{x}_1,\ldots,\vec{x}_j,\ldots,\vec{x}_k,\ldots,\vec{x}_N)=\pm \psi(t,\vec{x}_1,\ldots,\vec{x}_k,\ldots,\vec{x}_j,\ldots,\vec{x}_N)$. For the plus-sign you have bosons, for the minus-sign you have fermions. Relativistic QFT as used to formulate the standard model you can show that all particles with integer spin are necessarily bosons and all particles with half-integer spin are fermions.