Occasionally we have to carry out calculations with some effective Lagrangians
supplied by our particle physics friends (possibly related to new physics
beyond the standard model). For example, we could be given a Lagrangian density

where is some scalar field, is the Dirac field (electrons) and is a coupling constant. The Dirac equation that is conventionally used in atomic physics reads

(I suppress interactions of electrons with each other and with the nucleus). Given what is that extra operator that I would have to add to my Dirac Hamiltonian? I consistently derive in this tutorial (pdf).

For the impatient, the result is \begin{equation}
V^{\prime}\psi=-\gamma_{0}\left( \frac{\partial\mathcal{L}^{\prime}}
{\partial\bar{\psi}}-\partial_{\mu}\left( \frac{\partial\mathcal{L}^{\prime}
}{\partial\left( \partial_{\mu}\bar{\psi}\right) }\right) \right) .
\end{equation} Applications to axions and "Higgs portal" interactions are also covered in the tutorial (pdf).