Second Quantization for Fermions: Creation Operator

[jhosamelly]

So, I'm studying Second Quantization for fermions and came across this equation. I was just wondering why there is a summation needed? And why do we do it with (i≠p).? Please can someone explain this to me?

Reply and help is much appreciated.

 

[Darwin123]

So, I'm studying Second Quantization for fermions and came across this equation. I was just wondering why there is a summation needed? And why do we do it with (i≠p).? Please can someone explain this to me?

Reply and help is much appreciated.

This is basically a mathematical expression for the Pauli Exclusion Principle.
Two ways to state the Pauli exclusion principle are:
1) Two or more fermion particles can not share the same quantum state.
2) A specific quantum state can have either 0 or 1 fermion particles.

 

[jhosamelly]

Ow! Yes, I've read that! But why the need for Np? And why is -1 raised to Np if we just want to keep its sign?

 

[jhosamelly]

i'll post the whole page so you would have an idea about my question

So here, for those I put check on. Why is there a (-1)^Np? I know that's part of the creation and annihilation operator, but why is it there?

Second, the equation I encircled has Nq+1. How did that happen? Thanks.

 

[andrien]

It just arises because of the antisymmetry required,which introduces a factor of -1.

 

[jhosamelly]

Hmmm.. Because wavefunction for fermions are antisymmetric? Ok. Got that! But how was the encircled eqn derived? And the last equation is also giving me a hard time. I am trying to derive everything here.

 

[member 11137]

Hmmm.. Because wavefunction for fermions are antisymmetric? Ok. Got that! But how was the encircled eqn derived? And the last equation is also giving me a hard time. I am trying to derive everything here.

Hy, you seem to be lost.
Perhaps, this reference will be helpful:
www.scholarpedia.org/article/Second_quantization
(please respect the recommandations at the bottom of the page)
These operators seem to be spinors and have a typical anticommutative behavior.
I cannot help you directly, sorry.
Good luck