In high-energy electron diffraction calculations, it is common practice to make a small-angle or "paraxial" approximation, which involves dropping a second-order derivative term in the governing equation for the system. Assuming we have an electron plane wave in $z$ incident upon a specimen potential $U(\textbf{r})$, we start from the Schrödinger equation:
$$ \{\nabla^2 + 4\pi^2 [U(\textbf{r}) + K^2]\}\Psi(\textbf{r}) = 0$$
And assume the wave can be described as a modulated plane wave:
$$ \Psi(\textbf{r}) = \exp(2\pi i K z) \Phi(\textbf{r}) $$
This yields
$$ \frac{\partial^2 \Phi(\textbf{r})}{\partial z^2} + 4\pi i K \frac{\partial \Phi(\textbf{r})}{\partial z} + \Delta \Phi(\textbf{r}) + 4\pi^2 U(\textbf{r})\Phi(\textbf{r}) = 0$$
where $\Delta$ is the Laplacian in $x$ and $y$. At this point we argue that the second-order derivative in $z$ is small relative to the other terms, because most of the variation of $\Psi$ in $z$ is described by the $\exp(2\pi i K z)$ factor. So we discard that term and we're left with:
$$ 4\pi i K \frac{\partial \Phi(\textbf{r})}{\partial z} + \Delta \Phi(\textbf{r}) + 4\pi^2 U(\textbf{r})\Phi(\textbf{r}) = 0$$
My question is essentially about whether it is "proper" to just drop a derivative term like this. More specifically:
- Even if the dropped term is indeed small, it seems like it drastically alters the "character" of the differential equation. The order in $z$ has changed, the boundary conditions required to specify the solution are qualitatively different, and the solution method is much more straightforward (we can solve plane-by-plane in $z$). Is this really a valid way to solve the original differential equation?
- If this is indeed a "dodgy" mathematical procedure, is there a better way to make the approximation? And how can we understand why the "dodgy" procedure seems to give the (approximately) right answer? I assume some kind of series expansion could be made but I'm not sure how or where to do it.
- Is there some branch or theorem of mathematics that I can refer to to understand more about this type of problem? It's been a number of years since I studied calculus so I've forgotten how to think rigorously about these problems.