Why is the Yukawa potential equal to Green's function for free space? I recently saw the computation for Green's function for the Helmholtz equation
$$-\Delta f + \kappa^2 f=0$$
in free space. The computation is done using the Fourier transform, and it turns out that in the 3-dimensional case, Green's function is given by:
$$G(\kappa,r)=\frac{e^{-\kappa r}}{4\pi r}.$$
This reminded me very much of the Yukawa potential (or screened Coulomb potential) which you often see in models in electrodynamics or quantum mechanics.
I was wondering whether there is a simple heuristic to explain this result. For instance, Green's function for the Laplacian is given by the potential of a single particle (and this agrees with the case $\kappa=0$ which corresponds to no screening). The physical intuition for this is clear - you in a sense reproduce your solution by "integrating" over the effect of all point charges in space. But somehow the case $\kappa>0$ corresponds to the charged particle being screened, where the decay rate of the potential depends on the parameter $\kappa$. Is there a nice way to picture this?
Thanks in advance.
 A: Helmholtz equation is identical with the coordinate part of Klein-Gordon equation, which is the wave equation for a massive particle, so no wonder that they produce identical solutions.
Note also that Helmholtz equation does not imply screening: it is the equation obtained from the wave equation after separation of time and space variables. The equation that leads to Yukawa potential for a screen particle also has the form of Helmholtz equation, but it is really a linearized equation of Debye-Hückel theory, which is also referred to as screened Poisson equation.
Remarks

*

*If we want to be really precise, then Helmholtz equation is actually
$$
\Delta u(\mathbf{r}) + \kappa^2 u(\mathbf{r})=0,
$$
whereas the screened Poisson equation is
$$
\Delta u(\mathbf{r}) - \kappa^2 u(\mathbf{r})=-f(\mathbf{r}),
$$
that is the two differ by sign. But of course, sometimes we have to consider imaginary frequencies (decaying modes) of Helmholtz equation, in which $\kappa \rightarrow i\kappa$ and the two equations would look identical.

Klein-Gordon for time-independent case is
$$
\Delta\psi(\mathbf{r})-\frac{m^2c^2}{\hbar^2}\psi(\mathbf{r})=0
$$

*

*This is the most general second order partial differential equation that is invariant to arbitrary rotations in 3D.

*Linear screening could be obtained as the zero frequency limit of polarization propagator, i.e., as the exchange by virtual plasmons - in the same way as interactions are described in quantum electrodynamics.

