No more than two electrons in the entire universe can have the same momentum

[LostConjugate]

It is a bit more logical that no more than two electrons can occupy the same position in space, however the Pauli Exclusion principle also concludes that no state exists with more than two electrons having the same momentum. Even the state of the entire universe.

How is this physically logical?

If one electron on Earth gains some momentum, some other electron in another galaxy must change it's momentum?

 

[SpectraCat]

It is a bit more logical that no more than two electrons can occupy the same position in space, however the Pauli Exclusion principle also concludes that no state exists with more than two electrons having the same momentum. Even the state of the entire universe.

How is this physically logical?

If one electron on Earth gains some momentum, some other electron in another galaxy must change it's momentum?

I think the key point is that the Pauli exclusion principle is only concerned with electrons that are part of the same coherent wavefunction. In other words, if you have two H-atoms, A & B, separated by a distance of 1 meter, they can effectively be considered to have separate and independent wavefunctions. So it is no problem if the 1s electron in atom A has precisely the same quantum numbers as the 1s electron in atom B. However, if we move the two atoms close enough together so they start to interact, all of that changes. The interaction splits the energy levels that were degenerate at large distance, and you have to talk about the overall wavefunction of the system, and now the Pauli exclusion principle applies to the molecular states.

If you want to get into the more philosophical aspects of this, I think the answers probably lie in decoherence. The reason that two widely separated H-atoms can be considered as independent systems is that, even if they start out as part of the same system, as the separation between them increases, the amount of energy it takes to perturb the system and induce decoherence decreases until it is negligibly small. At that point, even vacuum fluctuations can break the coherence of the two-atom state. One that happens, the two atoms are independent systems and the PEP is no longer relevant.

 

[LostConjugate]

I think the key point is that the Pauli exclusion principle is only concerned with electrons that are part of the same coherent wavefunction. In other words, if you have two H-atoms, A & B, separated by a distance of 1 meter, they can effectively be considered to have separate and independent wavefunctions. So it is no problem if the 1s electron in atom A has precisely the same quantum numbers as the 1s electron in atom B. However, if we move the two atoms close enough together so they start to interact, all of that changes. The interaction splits the energy levels that were degenerate at large distance, and you have to talk about the overall wavefunction of the system, and now the Pauli exclusion principle applies to the molecular states.

If you want to get into the more philosophical aspects of this, I think the answers probably lie in decoherence. The reason that two widely separated H-atoms can be considered as independent systems is that, even if they start out as part of the same system, as the separation between them increases, the amount of energy it takes to perturb the system and induce decoherence decreases until it is negligibly small. At that point, even vacuum fluctuations can break the coherence of the two-atom state. One that happens, the two atoms are independent systems and the PEP is no longer relevant.

Point noted. Although the electrostatic force never reaches 0 over any distance. So in all fact, nothing is isolated by distance, there is still a contribution from the protons and electrons to the Hamiltonian. A single state can be constructed.

 

[SpectraCat]

Point noted. Although the electrostatic force never reaches 0 over any distance. So in all fact, nothing is isolated by distance, there is still a contribution from the protons and electrons to the Hamiltonian. A single state can be constructed.

The state can be constructed, you are correct .. however see my point about decoherence.

 

[Vanadium 50]

the Pauli Exclusion principle also concludes that no state exists with more than two electrons having the same momentum. Even the state of the entire universe.

This is not true.

Tha Pauli exclusion principle says that no two electrons can share (all) the exact same quantum numbers. This is a consequence of the spin-statistics theorem, which states that swapping any two electrons will cause the wavefunction to change sign.

 

[matonski]

If one electron on Earth gains some momentum, some other electron in another galaxy must change it's momentum?

By the uncertainty principle, if you know the momentum of an electron exactly, then you can't simultaneously say that its on earth.

On the other hand, if you know that one electron is on Earth and one electron is in another galaxy, then they are obviously in different states.

 

[LostConjugate]

This is not true.

Tha Pauli exclusion principle says that no two electrons can share (all) the exact same quantum numbers. This is a consequence of the spin-statistics theorem, which states that swapping any two electrons will cause the wavefunction to change sign.

If you put 2 electrons in a state, both with the same spin [polarity] and momentum, the wave equation is symmetric, therefore cannot exist in nature.

There is no other option than to have one electron spin up and one spin down which leaves no room for any other electrons.

By the uncertainty principle, if you know the momentum of an electron exactly, then you can't simultaneously say that its on earth.

On the other hand, if you know that one electron is on Earth and one electron is in another galaxy, then they are obviously in different states.

That is profound!

 

[Vanadium 50]

If you put 2 electrons in a state, both with the same spin [polarity] and momentum, the wave equation is symmetric, therefore cannot exist in nature.

No, that's false. Position matters as well.

 

[LostConjugate]

No, that's false. Position matters as well.

Plug any position in you would like for x_1 and x_2, the equation is still symmetric.

[tex] e^(\frac{ipx_1}{\hbar})e^(\frac{ipx_2}{\hbar})X_+X_+ [\tex]

 

[matonski]

That is profound!

I don't know if you are being sarcastic, but I do think this answers the question. If you know one electron is on Earth and the other electron is in another galaxy, then their position-space wavefunctions are obviously different. Therefore, their Fourier transforms (aka momentum-space wavefunctions) will also be different.

 

[LostConjugate]

I don't know if you are being sarcastic, but I do think this answers the question. If you know one electron is on Earth and the other electron is in another galaxy, then their position-space wavefunctions are obviously different. Therefore, their Fourier transforms (aka momentum-space wavefunctions) will also be different.

Opps I did not mean to sound sarcastic, what I mean't was that it was a great answer and made me think. I don't think it disproves that no more than two electrons can have the same momentum in the entire universe though.

 

[matonski]

Our classical ideas align more closely with expectation values anyway. It's definitely possible for any number of electrons in the universe to have the same expectation value of momentum.

 

[LostConjugate]

Our classical ideas align more closely with expectation values anyway. It's definitely possible for any number of electrons in the universe to have the same expectation value of momentum.

Another good point.