# Pauli Exclusion Principle, what does it say?

• Inquisiter
In summary, the Pauli exclusion principle states that no two fermions can be in the same quantum state. Even though there may be multiple eigenstates with the same energy, only a finite number of electrons can occupy them due to the principle. The number of states is limited by the number of independent states, similar to having a finite number of unknowns in algebra. It is not explicitly stated that the states must be orthogonal, but they must be independent. This is enough to expand in, similar to a Taylor expansion in non-orthogonal functions.
Inquisiter
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.

## 1. What is the Pauli Exclusion Principle?

The Pauli Exclusion Principle is a fundamental principle in quantum mechanics that states that no two identical fermions (particles with half-integer spin) can occupy the same quantum state simultaneously. This means that fermions cannot have the same set of quantum numbers, such as energy level, spin, and orbital angular momentum, in a given system.

## 2. Why is the Pauli Exclusion Principle important?

The Pauli Exclusion Principle is important because it explains the stability of atoms and the periodic table of elements. Without this principle, electrons would all occupy the lowest energy orbital, resulting in unstable and unpredictable chemical properties.

## 3. How was the Pauli Exclusion Principle discovered?

The Pauli Exclusion Principle was first proposed by Austrian physicist Wolfgang Pauli in 1925 as a solution to the anomalous behavior of electrons in atoms. It was later confirmed through experiments and is now considered one of the most fundamental principles in quantum mechanics.

## 4. Does the Pauli Exclusion Principle apply to all particles?

No, the Pauli Exclusion Principle only applies to fermions, which include particles such as electrons, protons, and neutrons. Bosons, on the other hand, do not follow this principle and can occupy the same quantum state simultaneously.

## 5. How does the Pauli Exclusion Principle relate to the electron configuration of atoms?

The Pauli Exclusion Principle determines the arrangement of electrons in an atom's orbitals. Since no two electrons can have the same set of quantum numbers, electrons fill orbitals in a specific order, starting with the lowest energy level and following the Aufbau principle and Hund's rule.

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