# Question about the Pauli exclusion principle.

## Main Question or Discussion Point

Suppose there are only two states, and that only two electrons could fit in them (spin states for example), but wouldn't these two states form a basis and so generate an infinite number of states that are linear combinations of these two, so three electrons could be in three different states.

Obviously thats wrong, but why? do they have to be in orthogonal states?

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The Pauli exclusion principle arises from the requirement that the wavefunction of the system be antisymmetric under the exchange of fermionic degrees of freedom. Now you may try to write down the wavefunction with three particles, but you'll find that the antisymmetry property causes such a wavefunction to vanish.

Suppose there are only two states, and that only two electrons could fit in them (spin states for example), but wouldn't these two states form a basis and so generate an infinite number of states that are linear combinations of these two, so three electrons could be in three different states.

Obviously thats wrong, but why? do they have to be in orthogonal states?

I think there is some confusion here. There are two states, yes, so in principle, you can form an infinite number of states through linear combinations, but those are one particle states.

For a two particle state, the only one allowed is the state where one particle is spin up and the other is spin down. There is only one state for for the combined system.

I hope that helps.

Should i forward a conclusion that linear combination of the spin functions of the electron cannot be done: that means ultimately there are only 2 possible spin states for an electron ! Anyone can further comment this ?

Should i forward a conclusion that linear combination of the spin functions of the electron cannot be done: that means ultimately there are only 2 possible spin states for an electron ! Anyone can further comment this ?
There are only two possible BASIS states for the spin states for an electron since they are spin 1/2. However, there an infinite number of spin states for an electron because you can make any number of other states by performing a linear combination of these 2 states.

I hope I got your question correct.