# Calculate sum. Fermionic operators

1. Jun 5, 2013

### LagrangeEuler

1. The problem statement, all variables and given/known data
How to calculate?
$\sum _{i,j} \langle 0|\prod_n \hat{C}_n \hat{C}^+_i\hat{C}_j \prod_n \hat{C}^+_n|0 \rangle$

2. Relevant equations
$\hat{C}^+, \hat{C}$ are fermionic operators.
$\{\hat{C}_i,\hat{C}^+_j\}=\delta_{i,j}$

3. The attempt at a solution
I have a question. What is $|0 \rangle$? Is that maybe $|0 0\rangle$?

2. Jun 5, 2013

### PhysicsGente

Are the $\hat{C}$'s ladder operators?

Last edited: Jun 5, 2013
3. Jun 5, 2013

### Mute

Presumably you have some number of Fermions, $N$. Then, $\left| 0 \right\rangle$ represents the vacuum state which contains no Fermions - you can take it as a shorthand for

$$\underbrace{\left| 0 0 \dots 0 \right\rangle}_{N~\rm{terms}}.$$

So, the Fermion creation operator for the kth fermion when acting on this state will create a Fermion in the kth slot of the vacuum state (changing the 0 to a 1).

That is,

$$\hat{c}_k^\dagger \left| 0 0 \dots 0_k \dots 0 \right\rangle = \left| 0 0 \dots 1_k \dots 0 \right\rangle,$$
where I used a subscript $k$ to denote that that is the kth entry in the state.

4. Jun 5, 2013

### LagrangeEuler

Tnx.

$c^+_k|00…0k…0\rangle=|00…1k…0\rangle$
and is maybe
$c^+_k|00…1k…0\rangle=0?$
Then
$\prod_n c^+_n |000...0\rangle=|111...1\rangle$

5. Jun 5, 2013

### LagrangeEuler

So result is
$\sum_{i,j}\langle 1111...|\hat{C}^+_i\hat{C}_j|111...\rangle=\sum_{i,j}\delta_{i,j}?$

6. Jun 5, 2013

### Mute

That's what it looks like to me. Note that since you're summing over i and j you should get a number out at the end.

7. Jun 6, 2013

### LagrangeEuler

And is it necessarily to $\langle 1...1|1...1 \rangle$ be $1$?

8. Jun 6, 2013

### sgd37

this is a problem in anti commutation. Anti commute the annihilation operators past all the other operators so that they may annihilate the vacuum, in other words you have to normal order the operators

9. Jun 8, 2013

### Mute

While not strictly necessary, typically the wavefunctions are defined to be normalized, no?