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

mcas
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Homework Statement
Using the following anticommutator relations of fermionic one-particle operators:
[itex]\{\hat{c}_{k\alpha},\hat{c}_{k'\beta} \}= \{ \hat{c}^\dagger_{k\alpha}, \hat{c}^\dagger_{k'\beta} \} = 0[/itex]
[itex]\{\hat{c}_{k\alpha},\hat{c}^\dagger_{k'\beta} \}=\delta_{kk'}\delta_{\alpha\beta}[/itex]

Show that the expected value for a vacuum state [itex]|\phi_0>[/itex] is:

[itex] <\phi_0| \hat{c}_{-k \downarrow} \hat{c}_{k \uparrow}\hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0>=1
[/itex]
Relevant Equations
Given in the homework statement
<br /> \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<br />

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!
 
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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.
 
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Gaussian97 said:
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!
 
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