Definition of fermionic annihilation operators

[malcomson]

Hi, I'm revising for an exam and I came across a past paper that has a question on annihilation operators, It asks what happens when acting on a wavefunction with a group of different creation/annhilation operators (all identical fermions..

It's quite simple apart from the fact that it includes both spin up and spin down options and in my notes I only have a simplified case of all spin up fermions.

My question is, if the annihilation operator a_i is has the constants (-1)^n₁+n₂+...+n_i-1 times n_i where n₁ etc are the occupancies of each state. Does this include both spin states of should I treat them as separate sets.

I think I should include both spins as that would give the anticommutation relations between operators of different spins, allowing antisymmetry of a function under the interchange of two particles of opposing spin but I'm not really confident about it.

Apologies if unclear, am yet to figure out putting equations in posts.

 

[BigJDubs]

Yes, you should include both spin states. The annihilation operator is a linear combination of all the single particle states with the coefficients being determined by the occupancy of each state. As such, it should include both spin up and spin down states in order to properly represent the system. This is also necessary to ensure that the anticommutation relations between operators of different spins hold, which are necessary for antisymmetry of a function under the interchange of two particles of opposing spin.