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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.

Roger V.
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GSofer
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    To me it is unclear what question I should actually try to answer... (e.g. compare the title to your last paragraph) The title question is more or less answered by: "That's the way Yukawa built his theory in the 30s." Or: "The Green function for the a certain equation was named Yukawa potential..." Anyway, the similarity between the Green function for Helmholtz's equation and the Yukawa potential is no surprise, the only difference between the potentials and corresponding equations is an imaginary factor $\kappa \rightarrow \text{i}\kappa$. – kricheli Nov 29 '22 at 08:57
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    @kricheli I am not aware of the history of how the Yukawa potential was originally developed. I came across two seemingly unrelated things which are described by the same function - the potential of a screened charge, and Green's function for a not directly related problem (the Helmholtz equation). The question is if there exists a physical intuition for this relation. Again, since I am not aware of the history, this might be immediate (and in this case feel free to write this as a complete answer). I am looking for an explanation similar to the one for the Laplacian described in my question. – GSofer Nov 29 '22 at 09:06
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    Note if we think in terms of either Helmholtz or Klein-Gordon viz. @RogerVadim's answer the Yukawa potential is specifically the spherically symmetric time-independent solution, i.e. it depends on $r$ but none of $\theta,,\varphi,,t$. – J.G. Nov 29 '22 at 11:12
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    You've probably learned about the derivation of the Coulomb potential out of the one-photon exchange amplitude; extend it to a one-pion-exchange amplitude. – Cosmas Zachos Nov 29 '22 at 15:34
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    @GSofer I see. Well, Roger Vadim's answer provides pretty much all the context you need. The answer to your question might be that it is simply a coincidence that the Helmholtz equation arising from problems of wave motion and the linearized Debye-Hückel problem describing charge screening yield the same results. I doubt that you can connect them via some physical intuition. – kricheli Nov 29 '22 at 16:59

1 Answers1

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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.
Roger V.
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