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I was trying to solve a statistical mechanics exercise and got stuck.

$\textbf{Attempt}$

The Hamiltonian matrix that describes the Quantum Gas is given by:

$$H =\begin{bmatrix} -\epsilon & t & 0 \\ t & -\epsilon & 0 \\ 0 & 0 & -2\epsilon \end{bmatrix}$$

From which, I got the following energies: $E_1 = -2\epsilon$, $E_2 = -\epsilon - t $ and $E_3 = -e + t$

Knowing that the partition function is given by: $Z = Tr[\exp(-\beta H)]$, thus the partition function is: $$Z = \exp(2\beta \epsilon) + \exp(\beta (\epsilon + t)) + \exp(\beta (\epsilon- t)) $$

Now, the calculation of the energy mean value seems straightforward:

$$\langle E \rangle = -\frac{\partial}{\partial \beta} \log(Z) \langle = \rangle \langle E \rangle = -\frac{2\epsilon \exp(\beta 2 \epsilon) + (\epsilon + t) \exp(\beta(\epsilon + t)) + (\epsilon - t) \exp(\beta (\epsilon - t))}{\exp(2\beta \epsilon) + \exp(\beta (\epsilon + t)) + \exp(\beta (\epsilon- t))} $$

However, when I try to plug some limits (like $\beta \epsilon \gg 1 $ or $\beta \epsilon \ll 1 $) to see what is the asymptotic behaviour of $\langle E \rangle$, I cannot get any conclusions.

What am I doing wrong?

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At low temperatures ($\beta\to\infty$) the partition function is dominated by the ground state and the average energy should be the ground state energy. This should be correct when taking the limit on the right.

At high temperatures ($\beta\to0$) all states are equally likely, and the average energy is simply the average energy of the states. Again, you can set $\beta=0$ and see that.

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