Pauli Exclusion Principle, what does it say?

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Discussion Overview

The discussion revolves around the Pauli Exclusion Principle and its implications for quantum states of fermions, particularly in the context of atomic structure and electron configurations. Participants explore the mathematical foundations of the principle, the nature of quantum states, and the effects of electron interactions in potential fields.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether the states related to the Pauli Exclusion Principle must be orthogonal or if non-orthogonal states can also be considered, raising concerns about the mathematics involved.
  • Another participant asserts that with three eigenstates, only three independent states can exist, suggesting that this limits the number of electrons to six.
  • A follow-up inquiry is made regarding the solutions to the Schrödinger equation for an electron in a field produced by multiple electrons versus a single nucleus, questioning if the energy levels change due to the presence of other electrons.
  • There is a discussion about the distinction between orthogonality and independence of states, with one participant noting that while orthogonal states simplify calculations, independence is sufficient for expansion purposes.

Areas of Agreement / Disagreement

Participants generally agree on the limitation of independent states in relation to the Pauli Exclusion Principle, but there is disagreement regarding the necessity of orthogonality versus independence in defining quantum states.

Contextual Notes

There are unresolved questions about the mathematical treatment of non-orthogonal states and the implications of electron interactions on energy levels, which depend on the definitions and assumptions made in the discussion.

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I know I already posted this question, but it seems to have gotten "lost" among the other questions in the same thread. This is really confusing me now, so I'll ask it again.

The pauli exclusion principle says that no two fermions can be in the same quantum state. But if we have three eigenstates (as we do for an atom for n=2, l=1), can't we produce an infinite number of states (each one with the same energy) by superimposing the three eigenstates(each time with different coefficients)? So when we talk about quantum states as they relate to the Pauli principle, must these states be orthogonal or what?? What is the mathematics of it? I sort of know how we write out the states as a matrix and take the determinant to asymmetrize the wavefunction. But can the indivifual states be non orthogonal? Why or why not? I did a Google search, but didn't find much useful info. As far as I know, we can place only 6 electrons in n=2, l=1 state, not an infinite number of electrons. And a follow up queston: for one electron in an electric field of a point charge (the nucleus) we get a bunch of bound states as solutions to the Schr. eq. But when the Schr. equation is solved for an electron moving in a field produced by the nucleus AND other electrons (say, using the effective potential method), do we get basically the same solutions (just shifted in energy) as if there were no other electrons, but just the nucleus? I mean, we don't get any extra energy levels because of the other electrons which contribute to the total potential, right?
 
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If there are only 3 estates, no linear combinations can give more than 3 INDEPENDENT states. The key word is independent. So you could put no more than 6 electrons in.
It is like having three unknowns in algebra. You can write many equations for the 3 unknowns, but only 3 will be independet.
For your followup: The number of stats does not depend on what is producing the potential.
 
Meir Achuz said:
If there are only 3 estates, no linear combinations can give more than 3 INDEPENDENT states. The key word is independent. So you could put no more than 6 electrons in.
It is like having three unknowns in algebra. You can write many equations for the 3 unknowns, but only 3 will be independet.
That's what I was thinking. But it's usually not stated explicitly that the states must be independent. Don't the states have to be orthogonal, not just linearly independent though?
 
"Orthogonal" makes them easier to use, but "independent" is enough to expand in.
For instance, a Taylor expansion is an expansion in non-orthogonal functions.
 

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