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Homework Statement
I have been given the Hamiltonian
[tex]H = \sum_{k} (\epsilon_k - \mu) c^{\dag} c_k[/tex]
where [tex]c_k[/tex] and [tex]c^{\dag}_k[/tex] are fermion annihilation and creation operators respectively. I need to calculate the ground state, the energy of the ground state [tex]E_0[/tex] and the derivative [tex]\frac{\delta E_0(\mu)}{\delta \mu}[/tex]. Apparently this last quantity is 'famous' and I should recognise it. However, I think that I am making some fundamental mistake quite early on.
Homework Equations
I know that
[tex]c^{\dag} c |1> = 1|1>[/tex]
and
[tex]c^{\dag}c|0>=0|0>[/tex]
So that
[tex]c^{\dag}|0> = |1>[/tex]
and
[tex]c|1> = |0>[/tex]
and
[tex]c|0>=0[/tex]
and
[tex]c^{\dag}|1> = 0[/tex]
(All of this is proven by writing these operators as matrices and multiplying by state vectors. These relations are confirmed in 'Quantum theory of solids' by Kittel)
The Attempt at a Solution
But when it comes to calculating the ground state of this Hamiltonian, I find something unusual..
[tex]H|0> = \sum_{k}\epsilon_k c_k^{\dag} c_k |0> - \mu \sum_k c_k^{\dag}c_k|0> \\
= \sum_k \epsilon_k|1> - \mu|1>
[/tex]
Using the first relation.
How do I now calculate the energy of this ground state?
[tex]<0|H|0> = <0|\sum_k \epsilon_k|1> - <0|\mu|1>[/tex]
What do I do with this? Have I made some fundamental error somewhere? This doesn't look right to me.