Pauli's Exclusion Principle  

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One very important concept and principle of quantum mechanics is the Pauli exclusion principle. Responsible for things such as different electron orbits, it can be observed in many places.

The exclusion principle states that no two Fermions can occupy the same quantum state. 

But what does this mean?

This can be shown in terms of the wavefunction of the fermions, and in particular the wavefunction of a system comprising two neutrons. The wavefunction of a fermion describes mathematically its position and other such quantum properties.

According to the Pauli exclusion principle the following is true about the wavefunction (Ψ) of a composite system of positions of two fermions: 

The formula above describes the fact that if the two neutrons are swapped, the composite wavefunction is multiplied my -1. However, if the wavefunction is the same for both:

However, the exclusion principle states that Ψ(n1,n2)= -Ψ(n2,n1). Thus for both the above to be true:

Therefore, it is mathematically impossible for 2 fermions to have the same wavefunction, and therefore occupy the same quantum state. Unless one (or both) the wave functions are equal to zero (meaning that they don't exist).

I know the proof above is very basic, so I would love to learn how to properly prove it mathematically, I’m guessing something to do with the Schrödinger equation will help me? 

PlanckTime - Amin

 
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A lot of proofs of this are wrong, but yours is sweet and simple. The meaning of the Pauli Exclusion is formally told as that the particles with half-integer spins, or fermions, are able to be described by anti-symmetric wave functions, or a wave function that has (a,b) and not (b,a), and particles with an integer spin or bosons, will be described be a symmetric wave function, (a,b) and (b,a) are there. I described it quite simply here, and I am sure that you know spins and the asymmetric relation, but I just wanted to be more specific in describing it.

That one guy that's always on

 
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