Laplaces' s equation in spherical coordinates

[mjordan2nd]

After setting up Laplace's equation in spherical coordinates and separating the variables, it is not clear to me why the constants are put in the form of l(l+1) and why m runs from -l to l. Could anyone please help me ununderstand, or better yet, point me to a source that explains the entire process of separating Laplace's equation in spherical coordinates in detail? I have not been able to find anything on the internet whi h clarifies these points. It merely states them as facts.

 

[DEvens]

There is a quantum mechanics class in your immediate vicinity. You'd better have some light or you will get eaten by a Green function.

But seriously... They are put in that form because people found solutions that have important applications, and it was very convenient that way. You are going to find angular momentum quantum numbers popping out at you any second now, just as one example.

 

[mjordan2nd]

At this stage I'm trying to solve electromagnetism problems where recasting Laplace's equation into spherical coordinates seems useful. I am trying to use a series solution for the polar part. I have seen the quantum argumemt using operators to get the result, but I would think it should pop out of the series solution itself, especially the constraints on m. However, if that is the case I can't see it.

 

[DEvens]

In E&M you will be seeing multi-pole expansion. And that is going to show up in the Laplace equation as well, also with the l and m values in the general scheme you have seen.

Keep going. It will become obvious why it is done that way.

 

[fluidistic]

After setting up Laplace's equation in spherical coordinates and separating the variables, it is not clear to me why the constants are put in the form of l(l+1) and why m runs from -l to l. Could anyone please help me ununderstand, or better yet, point me to a source that explains the entire process of separating Laplace's equation in spherical coordinates in detail? I have not been able to find anything on the internet whi h clarifies these points. It merely states them as facts.

You'll find the answer to why the constant is generally chosen as l(l+1) in http://www.luc.edu/faculty/dslavsk/courses/phys301/classnotes/laplacesequation.pdf