In quantum treatments of solids it is common to start off discussions by writing down the "full" first-principles Hamiltonian for a group of electrons and nuclei as

$$H = \sum_i \frac{\hat{p}_i^2}{2m_i}+\frac{e^2}{2}\sum_{i,j} \frac{Z_i Z_j}{\vert \mathbf{r}_i - \mathbf{r}_j \vert}$$

This Hamiltonian can in principle be solved for a many body wavefunction $\Psi(\mathbf{r}_1,\dots, \mathbf{r}_N)$ and contains all the physics of a solid.

There are relativistic corrections that can be made to this Hamiltonian. The simplest is replacing the kinetic energy term with higher order terms in $p$. But there are also spin-dependent terms (like the Zeeman term), and spin orbit coupling as discussed in this question.

Let's assume we still want to stay in the comfort of non-relativistic quantum mechanics (particle number is conserved, fixed Coulomb gauge, etc.), but want a more accurate first principles Hamiltonian.

My question is: what is the correct expression for the many-body Hamiltonian of a group of electrons/nuclei that includes all important relavistic corrections to leading order?

If you let me rephrase that question in a more hand-wavy way, how do we explicitly include relavistic corrections (Zeeman spin term, spin-orbit, Darwin, etc.) into the full many body Hamiltonian above to make sure it doesn't leave out important SR physics?

The reason I ask this question is that it is common in solids state texts to include relavistic corrections at the single particle level, but rarely at the level of the "full" many body treatment.


For a truly relativistic treatment you should start with the Dirac equation and add the periodic lattice potential - it is just as simple, as adding a potential to the Schrödinger equation, as the first equation suggests. It is a good exercise, but it is never done in practice, because band structure calculations are a science in themselves - relativistic treatment is really a minor complication in comparison. Therefore the spin-orbit terms (Rashba, Dresselhaus-D'yakonov-Perel, Elliot-Yafet, etc.) are usually posited on the symmetry (lattice and relativistic) grounds and their constants are determined empirically.

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  • $\begingroup$ Of course you can just write out the Dirac equation, but that is not my question. That approach has the added hassle of negative energy (antiparticle) states that are irrelevant, so one would always take a non-relativistic approximation anyways. My question is also not about band structure, but the structure of the "first principles" Hamiltonian. $\endgroup$ – KF Gauss Mar 15 at 9:22
  • $\begingroup$ I should also point out that this Hamiltonian doesn't make any assumptions about the existence of a crystal lattice, it is just a many-body collection of electrons and nuclei with all particles treated equally. $\endgroup$ – KF Gauss Mar 15 at 9:25
  • $\begingroup$ Indeed, but this is how one derives relativistic corrections - by starting from the fully relativistic model rather than from the fully non-relativistic one. For a hydrogen atom these are usually worked through in any book treating Dirac equation. $\endgroup$ – Vadim Mar 15 at 9:25
  • $\begingroup$ I agree, so let me say it differently, my question is asking for the "worked out" non relativistic expression analogous to the hydrogen atom but for a many body system. $\endgroup$ – KF Gauss Mar 15 at 9:27
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    $\begingroup$ Then it should be very easy for you to add to your answer $\endgroup$ – KF Gauss Mar 15 at 9:39

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