- #1
Xyius
- 501
- 4
Problem Question
My question isn't an entire homework problem, but rather for a certain mathematical step in the problem which I assume to be very simple.
The problem is dealing with interacting fermion systems using second quantization formulas. I am essentially following my notes from class for this problem and the part I am stuck on is this.
<0,0|a_{2 \uparrow} a_{1 \uparrow}a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}a_{1 \uparrow}a_{2 \uparrow}a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=-1
My question is, why is this equal to -1?
Attempt at Solution
Here is my logic on how to evaluate this.
a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=|\uparrow \uparrow>
then
a_{1 \uparrow}a_{2 \uparrow}|\uparrow \uparrow>=|0,0>
then
a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=|\uparrow \uparrow>
finally
a_{2 \uparrow} a_{1 \uparrow}|\uparrow \uparrow>=|0,0>
Thus all together I get 1, not -1. Why is this equal to -1?
My question isn't an entire homework problem, but rather for a certain mathematical step in the problem which I assume to be very simple.
The problem is dealing with interacting fermion systems using second quantization formulas. I am essentially following my notes from class for this problem and the part I am stuck on is this.
<0,0|a_{2 \uparrow} a_{1 \uparrow}a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}a_{1 \uparrow}a_{2 \uparrow}a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=-1
My question is, why is this equal to -1?
Attempt at Solution
Here is my logic on how to evaluate this.
a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=|\uparrow \uparrow>
then
a_{1 \uparrow}a_{2 \uparrow}|\uparrow \uparrow>=|0,0>
then
a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=|\uparrow \uparrow>
finally
a_{2 \uparrow} a_{1 \uparrow}|\uparrow \uparrow>=|0,0>
Thus all together I get 1, not -1. Why is this equal to -1?