Pauli's exclusion principle

It means two identical fermions (say, any two electrons) cannot be in exactly the same state. So if you have a discrete way of labeling the states, say by atomic quantum numbers, then you can only have one electron in each of those states (or two if you are not counting the spin in the quantum numbers). One reason this is really important is that it gives us the periodic table-- atoms get a lot larger when you "break into a new valence shell" as you add electrons, and that's only because the new electrons can't go into the states that the previous electrons are already in.

 

Are you familiar with the harmonic oscillator?

Fermionic creation and annihilation operators are constructed using anti-commutators (instead of commutators

[tex]\{b_s,b^\dagger_{s^\prime}\} = \delta_{s\,s^\prime}[/tex]

[tex]\{b_s,b_{s^\prime}\} = \{b^\dagger_s,b^\dagger_{s^\prime}\} = 0[/tex]

The last equation is important b/c for s=s' it can be rewritten as :

[tex](b^\dagger_s)^2 = 0[/tex]

s,s' represent the quantum numbers

You can construct a states where each s, s', s'', ... is occupied by either zero or one particle. But if you want to construct a state with two particles sitting in state s you get

[tex]|2s\rangle = (b^\dagger_s)^2|0\rangle = 0[/tex]

due to the anticommutator.

 

They can be in the same state as long as they have different positions though right?

 

Sorry, I forgot to explain that creation and annihilation operators act in momentum space and that this means that s is a collection of momentum, spin, isospin, etc.}. Therefore they are not located anywhere in position space.