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Example 3.9 (from Introduction to Many-Body Physics)

(a) Enumerate the energy eigenstates of a single fermion Hamiltonian $H = Ec^{\dagger} c$

(b) Calculate the number of fermions at temperature T.

Solution

(a) The states are: $|0>$ and $|1>$ with energies $E_0 = 0, E_1 = E$.

(b) The number of fermions at temperature $T$ is given by

$$ <\hat{n}> = \operatorname{Tr} [\hat{\rho} \hat{n}], $$ where $\hat{n} = c^{\dagger}c$,

$$ \rho = e^{-\beta (\hat{H} - \mu \hat{N})}/Z $$ is the density matrix, and

$$ Z = \operatorname{Tr}[e^{-\beta (\hat{H} - \mu \hat{N})}] $$

is the partition function. And then he writes operators explicitly as matrices for this example:

$$ e^{-\beta (\hat{H} - \mu \hat{N})} = > \operatorname{diag}(1,e^{-\beta ({H} - \mu )}), \hat{n} = > \operatorname{diag}(0,1)... $$

from which the Fermi-dirac function for $<\hat{n}>$ comes straightforward.

To my shame I found out I don't understand some things:

  • it would make sense if $\hat{n} \equiv \hat{N}$ though in text it is not mentioned. Is it?

  • if $\hat{n}$ is a matrix then $c^{\dagger}$ is a matrix (or a vector) itself, but if $|0>$ and $|1>$ are wavefunctions from Fock space how can

$$ |1> = c^{\dagger} |0> $$ this equation hold?

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  • it would make sense if $\hat n \equiv \hat N$ though in text it is not mentioned. Is it?

Yes, in this particular case, since we're talking about a one-state system. But in general there could be many states and fermions in a system, and $n_k$ represents the occupation number operator at state $k$th and $N$ is the total number of fermion operator.

  • if n̂ is a matrix then c† is a matrix (or a vector) itself, but if |0> and |1> are wavefunctions from Fock space how can $$|1> = c^{\dagger} |0>$$ this equation hold?

In general, $n$ and $c^\dagger$ are just operators and $|0>$ and $|1>$ are just states. They can be represented in many ways. Using matrices and vectors is just one of the ways to represent them. For instance, we can use $ \left[ {\begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} } \right]$ to represent $n$ and $ \left[ {\begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} } \right]$ to represent $c^\dagger$, from here we have vectors $ \left[ {\begin{array}{c} 0 \\ 1 \\ \end{array} } \right] $ and $ \left[ {\begin{array}{c} 1 \\ 0 \\ \end{array} } \right] $ to represent $|0>$ and $|1>$, respectively. But note that these matrices and vectors are not unique!

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