Solving Laplace's Equation with Mass

[homology]

Hi folks, I'm drawing a blank on the following:

$$\frac{1}{r^2}\frac{\partial}{\partial r}r^2\frac{\partial \phi}{\partial r}+\mu^2 \phi=0$$

I 'know' the solutions which are linear combinations of $$e^{\pm\mu r}/r$$ as you can check, but I've been trying to see if I could show the fact (as opposed to just checking that it works, which it does).

Context: The above, as I'm sure you realize, is 'almost' Laplace's equation. "If" a photon had mass (the term $$\mu$$ would be proportional to it) the above equation would result for a region with no charge.

I'm guessing that the answer is simple so please be vague with your hints (I suppose that's almost asking for a smartass comment) as I'd like to figure it out myself.

Thanks

 

[AlephZero]

Expand the differentials so you have a 2nd order equation.

Then substitute $\psi = r \phi$

 

[homology]

Expand the differentials so you have a 2nd order equation.

Then substitute $\psi = r \phi$

awesome! was that inspired by experience or a general rule of thumb?

 

[AlephZero]

You said the solution was something like $e^{kr}/r$. I know how to solve a DE with solutions like $e^{kr}$ so factoring out the r seemed like a good thing to try

In cylindrical and spherical coordinates, solutions often have a factor of 1/r or 1/r^2 compared with cartesian coordinates, because they represent something being spread out over a plane or solid angle, compared with a strip of constant width.

The DEs often have the form 1/r^n d/dr (r^n times something) as well.

 

[blue_raver22]

I find this question intriguing and it raises some interesting concepts to consider. In general, Laplace's equation is used to describe the behavior of a scalar field in a region with no sources or sinks. In this case, the addition of the term \mu^2 \phi adds a mass term to the equation, which can be interpreted as the presence of a source or sink of mass in the region.

One possible approach to solving this equation would be to use separation of variables, where we assume that the solution can be written as a product of two functions, one that only depends on r (the radial coordinate) and one that only depends on \theta and \phi (the angular coordinates). This would give us a system of two ordinary differential equations, one for each function, which could then be solved using standard techniques.

Another approach could be to use the method of Green's functions, where we can write the solution as an integral over the region of interest. This method can be particularly useful for solving equations with non-homogeneous terms, such as the \mu^2 \phi term in this case.

In terms of showing that the solutions are linear combinations of e^{\pm\mu r}/r, we could use a combination of these approaches. By assuming a solution of the form e^{\mu r} for the first term, we can find the corresponding solution for the second term using separation of variables. This would then give us the general solution in the form of a linear combination of e^{\pm\mu r}/r.

Overall, the addition of a mass term in Laplace's equation adds an interesting twist to the problem and requires some additional techniques for solving. I would suggest exploring these techniques further and considering the implications of a mass term in the context of the physical system being studied.