Say I have a Fock space H with basis K={|k⟩|k∈N}. Then I consider the following single particle states:
|A⟩=∑k∈Kak|k⟩, |B⟩=∑k∈Kbk|k⟩.
I know that |k1k2⟩=1√2(|k1⟩⊗|k2⟩−|k2⟩⊗|k1⟩) is a valid fermionic two-particle state. I expected I could calculate the two particle state which contains particles A and B as
|AB⟩?=1√2(|A⟩⊗|B⟩−|B⟩⊗|A⟩).
But it turns out that
1√2(|A⟩⊗|B⟩−|B⟩⊗|A⟩)=1√2∑k1,k2∈K(ak1bk2|k1⟩⊗|k2⟩−ak2bk1|k2⟩⊗|k1⟩)=0.
So how do I write this two particle state |AB⟩? It should be expressible as
|AB⟩=∑k1,k2∈Kk1<k2ck1k2|k1k2⟩,
but what is ck1k2? Is it ck1k2=ak1bk2? Why?
Answer
Why should
1√2∑k1,k2∈K(ak1bk2|k1⟩⊗|k2⟩−ak1bk2|k2⟩⊗|k1⟩)=0
be true?
By switching the indices you get
1√2∑k1,k2∈K(ak1bk2|k1⟩⊗|k2⟩−ak1bk2|k2⟩⊗|k1⟩)=1√2∑k1,k2∈K(ak1bk2−ak2bk1)|k1⟩⊗|k2⟩.
As you can see, the symmetric terms vanish, but the antisymmetric ones remain. From this equation, it is easy to see that
ck1,k2=1√2(ak1bk2−ak2bk1).
Further information is given here.
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