Converting Laplacian to spherical coordinates.

In summary, the conversation is about converting the laplacian operator from Cartesian to spherical coordinates in the study of quantum chemistry. The person is looking for an easier method than the "brute-force" approach of using the multivariable chain rule and evaluating partial derivatives. They mention taking div(grad f) and converting standard basis vectors from Cartesian to spherical coordinates, but are not patient enough to go through the process. They are seeking an intermediate theorem that simplifies the conversion process and is not too theoretical for a freshman in college. A source is mentioned that does the brute-force method, but the person is looking for a higher-level simplification.
  • #1
scorpion990
86
0
Hey! I'm self-studying a bit of quantum chemistry this summer. My introductory P.chem book (David Ball) doesn't specifically show the conversion of the laplacian operator from Cartesian to spherical coordinates. I don't really feel satisfied until I've actually derived it myself... So... Question:

Is there an easier, non "brute-force" method of converting the laplacian from Cartesian to spherical coordinates? I know that I can rewrite it using the multivariable chain rule and evaluating dozens of partial derivatives, but I'd rather not. From my experience, there has to be an intermediate theorem which shortens the process.

I'd prefer something that is not too overly theoretical (I'm still a freshman!) I've taken undergraduate math up until differential equations, and I'm familiar with some elements of applied math.

Thanks!
 
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  • #2
See this wiki page: http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates"
 
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  • #3
That doesn't seem to show how to derive the formulas...
I started the conversion, but I'm not patient enough to go through will all of it... I guess I could take div(grad f), but then I'll need to know how to convert the standard basis vectors from cartesian into spherical coordinates. Err..

EDIT: I found a source which does the brute-force way:
http://planetmath.org/encyclopedia/DerivationOfTheLaplacianFromRectangularToSphericalCoordinates.html#foot1096

...and I'm not going to bother following the derivation step by step. There must be a higher-level simplification.
 
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1. What is the purpose of converting Laplacian to spherical coordinates?

The purpose of converting Laplacian to spherical coordinates is to simplify the calculation and analysis of functions in three-dimensional space. Spherical coordinates are particularly useful for problems involving spherical symmetry, such as in physics and engineering.

2. What is the formula for converting Laplacian to spherical coordinates?

The formula for converting Laplacian to spherical coordinates is:
∇²f = (1/r²) (∂/∂r)(r²∂f/∂r) + (1/r²sinθ)(∂/∂θ)(sinθ∂f/∂θ) + (1/r²sin²θ)(∂²f/∂φ²)

3. How do you visualize spherical coordinates?

Spherical coordinates can be visualized by imagining a spherical coordinate system with a fixed point at the origin, a radial distance from the origin represented by "r", an angle from the positive z-axis represented by "θ", and an angle from the positive x-axis on the xy-plane represented by "φ". The point in three-dimensional space is then located by the values of r, θ, and φ.

4. What are the advantages of using spherical coordinates over Cartesian coordinates?

One advantage of using spherical coordinates is that they are better suited for problems involving spherical symmetry. They also simplify the calculation and analysis of functions in three-dimensional space, especially for problems involving rotation and circular motion. Additionally, spherical coordinates can be useful for visualizing and understanding physical phenomena such as electromagnetic fields and fluid flow.

5. How do you convert a Laplacian equation in Cartesian coordinates to spherical coordinates?

To convert a Laplacian equation in Cartesian coordinates to spherical coordinates, you can use the formula mentioned in question 2. You will need to substitute the values of r, θ, and φ into the equation and then solve for the Laplacian (∇²) in terms of these spherical coordinates. It is also important to note that the unit vectors in spherical coordinates (eᵣ, eθ, and eφ) must be taken into account when converting the equation.

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