Laplaces' s equation in spherical coordinates

In summary, Laplace's equation is solved in spherical coordinates by separating the variables and putting the constants in the form of l(l+1) and m running from -l to l. The reason for this is that people found solutions that had important applications, and it was convenient to do so.
  • #1
mjordan2nd
177
1
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.
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
mjordan2nd said:
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
 

Related to Laplaces' s equation in spherical coordinates

1. What is Laplace's equation in spherical coordinates?

Laplace's equation in spherical coordinates is a partial differential equation that describes the distribution of a scalar quantity in a three-dimensional space. It is used to model physical phenomena such as heat transfer, electrostatics, and fluid flow.

2. What are the main applications of Laplace's equation in spherical coordinates?

Laplace's equation in spherical coordinates is used in various fields of science and engineering, including physics, mathematics, and engineering. It is commonly used to solve problems related to heat transfer, electrostatics, and fluid flow.

3. What are the boundary conditions for Laplace's equation in spherical coordinates?

The boundary conditions for Laplace's equation in spherical coordinates depend on the specific problem being solved. However, in general, the boundary conditions specify the values of the scalar quantity at the boundaries of the three-dimensional space.

4. How is Laplace's equation in spherical coordinates solved?

Laplace's equation in spherical coordinates can be solved analytically or numerically. Analytical solutions involve using mathematical techniques such as separation of variables, while numerical solutions involve using computational methods to approximate the solution.

5. What are some common simplifications made when using Laplace's equation in spherical coordinates?

Some common simplifications made when using Laplace's equation in spherical coordinates include assuming certain symmetries in the problem, such as axisymmetry or symmetry about the origin. Additionally, certain boundary conditions may be assumed to simplify the problem, such as assuming a constant value at the boundaries.

Similar threads

A Separation of variables is possible only in 11 coordinate systems?
    • Differential Equations
Replies
7
Views
2K
Solving Laplace's equation in spherical coordinates
    • Differential Equations
Replies
5
Views
3K
How to Solve the Laplace Equation for Potential Flow Around a Sphere?
    • Advanced Physics Homework Help
Replies
1
Views
916
I PDEs: Laplace's Equation over a Parallelogram
    • Differential Equations
Replies
10
Views
3K
A Laplace transform in spherical coordinates
    • Calculus
Replies
3
Views
2K
Find the field inside and outside a spherical geometry
    • Advanced Physics Homework Help
Replies
2
Views
567
A Laplace equation- variable domain
    • Differential Equations
Replies
2
Views
2K
I Laplace equation boundary conditions
    • Differential Equations
Replies
6
Views
1K
I Dot product of two vector operators in unusual coordinates
    • Quantum Physics
Replies
14
Views
1K
Laplace in Spherical and Cylindrical Coordinates
    • Calculus and Beyond Homework Help
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top