# Quantum statistical mechanics formalism

How do we solve a Hamiltonian written in second quantization by using quantum statistical formalism? For example, the following Hamiltonian

$$H = \sum_{i=1}^L c_{i+1}^\dagger c_i + h.c.$$

I have solved few examples in which $$H$$ had some function form but I am not sure how to find ground-state properties of Hamiltonian wirrten in second quantization. Especially, how do we solve this kind of Hamiltonian in computer? i.e. by using exact diagonalization (for small system sizes)

I am sorry of this question seems stupid. I couldn't find any article which explain things like this.

You should find the answer to your question in any basic reference which deals with the tight-binding model. You introduce the Fourier transforms of the annihilation and creation operaors. This diagonalizes the Hamiltonian in $$k$$-space. The energy associated to $$k$$ is the Fourier transform of the hoppings (1 in your case, in 1D, so just a cosine). See similar question here Dispersion relation in tight binding model with even indices only