- #1
tylerscott
- 28
- 0
Homework Statement
Homework Equations
and attempt at solutionI think I got the ground state, which can be expressed as [itex]|\Psi \rangle = \prod_{k}^{N}\hat{a}_{k}^{\dagger} |0 \rangle[/itex] .
Then for the density matrix I used:
[itex]\langle 0|\prod_{k'}^{N}\hat{a}_{k}\hat{a}_{k}^{\dagger}\hat{a}_{l}\prod_{k'}^{N}\hat{a}_{k'}^{\dagger} |0 \rangle[/itex].
Due to the commutation relations for fermions:
[itex]\hat{a}_{l}\prod_{k}^{N}\hat{a}_{k}^{\dagger} |0 \rangle=\left ( -1 \right )^{\sum_{i=1}^{l-1}}\prod_{k\neq l}^{N}\hat{a}_{k}^{\dagger} |0 \rangle[/itex]
So the density matrix becomes:
[itex]\langle 0|\left ( -1 \right )^{\sum_{i=1}^{l-1}+{\sum_{i=1}^{k-1}}}\prod_{k'\neq k}^{N}\hat{a}_{k}\prod_{k'\neq l}^{N}\hat{a}_{k'}^{\dagger} |0 \rangle[/itex].
I honestly don’t know what to do past this, or if I’m even on the right track. Any help is appreciated.
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