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I'm trying to figure out how one would write down the Hamiltonian of a free fermion system (eventually in second quantization) on a one dimensional lattice and I'm having trouble both coming up with anything myself and finding relevant material.

My "guess / gut feeling / starting point" was that I could write down the creation and annihilation operators for momentum space (which I guess I just assumed to be quantized in the same way that it would be for a continuous one dimensional box) and then figure out how to express those in terms of the creation and annihilation operators for real space. (I'm pretty shaky on fourier transforms so I wasn't sure I did that part correctly either) but I don't have a lot of justification for what I'm doing.

I've also spent time looking for relevant materials all over the internet but mostly I'm just finding results that are connected to things like Brillouin Zones, Bloch Waves and crystals. Should I be trying to learn about solid state physics to answer my question or would that be going in the wrong direction?

I would appreciate both direct responses and reading suggestions.

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    $\begingroup$ The simplest is nearest-neighbor hopping: $-t\sum_i(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i)$. $\endgroup$
    – Meng Cheng
    Commented Jun 25, 2015 at 2:10
  • $\begingroup$ I was hoping for something that would be equivalent to the momentum operator in the continuum case in the limit of many many lattice sites. $\endgroup$
    – Bill Huggins
    Commented Jun 25, 2015 at 21:55
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    $\begingroup$ This is equivalent to the momentum operator in the continuum if you take the continuum limit. $\endgroup$
    – Meng Cheng
    Commented Jun 25, 2015 at 23:54
  • $\begingroup$ Ah, thank you for your help! I was confused about how it should look and had ruled this out in my head. After reading your answer and thinking about it I was able to figure out where I was going wrong. $\endgroup$
    – Bill Huggins
    Commented Jun 26, 2015 at 23:04

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If the fermions are Dirac Fermions then you can use the Hamiltonian density listed on Wikipedia:$$\mathcal{H} = \Pi \gamma^0 \left[\vec{\gamma}\cdot \nabla + m\right]\psi,$$ where $\Pi$ is the field canonically conjugate to $\psi$. This is equation 7.5.37 from Weinberg's Quantum Theory of Fields, Vol I. Although, figuring out how to represent the gamma matrices when there is $1$ spatial dimension instead of $3$ is, if I recall correctly, not trivial, the form should still be the same.

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