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 
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).