mcas
- 22
- 5
- 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!
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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