Show that an expected value of a vacuum state is equal to 1

[mcas]

Homework Statement

Using the following anticommutator relations of fermionic one-particle operators:
$\{\hat{c}_{k\alpha},\hat{c}_{k'\beta} \}= \{ \hat{c}^\dagger_{k\alpha}, \hat{c}^\dagger_{k'\beta} \} = 0$
$\{\hat{c}_{k\alpha},\hat{c}^\dagger_{k'\beta} \}=\delta_{kk'}\delta_{\alpha\beta}$

Show that the expected value for a vacuum state $|\phi_0>$ is:

$<\phi_0| \hat{c}_{-k \downarrow} \hat{c}_{k \uparrow}\hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0>=1$

Relevant Equations

Given in the homework statement

$\langle \phi_0| \hat{c}_{-k \downarrow} \hat{c}_{k \uparrow}\hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| - \hat{c}_{k \uparrow} \hat{c}_{-k \downarrow} \hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| \hat{c}_{k \uparrow} \hat{c}^\dagger_{k \uparrow} \hat{c}_{-k \downarrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0|(1- \hat{c}^\dagger_{k \uparrow} \hat{c}_{k \uparrow} ) \hat{c}_{-k \downarrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0|\hat{c}_{-k \downarrow}\hat{c}_{-k \downarrow}|\phi_0\rangle - \langle \phi_0|\hat{c}^\dagger_{k \uparrow} \hat{c}_{k \uparrow} \hat{c}_{-k \downarrow}\hat{c}_{-k \downarrow}|\phi_0\rangle$

Then I changed $\hat{c}^\dagger_{k \uparrow} \hat{c}_{k \uparrow}$ in the second term to $(1- \hat{c}^\dagger_{k \uparrow} \hat{c}_{k \uparrow} )$ and the result was $\langle \phi_0| \hat{c}_{-k \downarrow} \hat{c}_{k \uparrow}\hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle$ which is exactly what I sarted from.

I don't know where to go from this so I would really appreciate any help!

 

[Gaussian97]

Where did you get this problem from?

Are the $\hat{c}_{k\alpha}$ usual annihilation operators?
How is the $\ket{\phi_0}$ state defined?

If $\ket{\phi_0}$ is the usual vacuum defined from the $\hat{c}$ operators then, if I'm not wrong, by definition $\hat{c}_{-k \downarrow} \ket{\phi_0}=0$ and the whole expectation value vanishes. So, either I'm confused by something, or the problem is wrong, or you have extra information that is crucial for solving the problem.

 

[mcas]

Where did you get this problem from?

Are the $\hat{c}_{k\alpha}$ usual annihilation operators?
How is the $\ket{\phi_0}$ state defined?

If $\ket{\phi_0}$ is the usual vacuum defined from the $\hat{c}$ operators then, if I'm not wrong, by definition $\hat{c}_{-k \downarrow} \ket{\phi_0}=0$ and the whole expectation value vanishes. So, either I'm confused by something, or the problem is wrong, or you have extra information that is crucial for solving the problem.

It's a problem formulated by my professor.
After you post I contacted him and indeed, he made a mistake in the homework statement, as the last operator $\hat{c}_{-k \downarrow}$ should be conjugated for the expected value to be one. In the original statement, the value will be 0.
Thank you!