# Transforming Cartesian Coordinates in terms of Spherical Harmonics

• I
• Athenian
In summary, Cartesian coordinates can be transformed into spherical harmonics by using the explicit expressions for the spherical harmonics and the Condon-Shortley phase convention.
Athenian
TL;DR Summary
How one could transform Cartesian coordinates (e.g. ##x##) in terms of spherical harmonics. Refer to the full post for more information.
As the subject title states, I am wondering how would one go about transforming Cartesian coordinates in terms of spherical harmonics.

To my understanding, cartesian coordinates can be transformed into spherical coordinates as shown below.

$$x=\rho \sin \phi \cos \theta$$

$$y= \rho \sin \phi \sin \theta$$

$$z= \rho \cos \phi$$

And, considering that I can take any function that is in terms of spherical coordinates as a sum over spherical harmonics, I am assuming that I can take any Cartesian coordinate value and transform it in terms of spherical harmonics.

That said, how would one go about doing that? For example, how would one take the Cartesian term ##x## and transform it in terms of spherical harmonics (perhaps with coefficients with it)? I tried to search online for the process to accomplish that but could find none.

Therefore, if anybody here could help me answer this question, I would greatly appreciate it. Thank you!

From the explicit expressions for the spherical harmonics it is straightforward to derive that $$x = \rho \sqrt{\frac{2\pi}{3}}(Y_{1,-1}(\theta, \phi) - Y_{1,1}(\theta, \phi)),$$
$$y = i\rho \sqrt{\frac{2\pi}{3}}(Y_{1,-1}(\theta, \phi) + Y_{1,1}(\theta, \phi)),$$
$$z = \rho \sqrt{\frac{4\pi}{3}}Y_{1,0}(\theta, \phi) ,$$
where I used the Condon-Shortley phase convention.

BvU and Athenian

## 1) What are Cartesian coordinates?

Cartesian coordinates are a system of coordinates used to describe the position of a point in space. They consist of three perpendicular axes (x, y, and z) and measure the distance of a point from the origin along each axis.

## 2) What are spherical harmonics?

Spherical harmonics are mathematical functions that describe the shape of a three-dimensional object in terms of spherical coordinates. They are commonly used in physics and engineering to represent the solutions to certain differential equations.

## 3) How do you transform Cartesian coordinates into spherical harmonics?

To transform Cartesian coordinates into spherical harmonics, you can use the following equations:

r = √(x² + y² + z²)

θ = arccos(z/r)

φ = arctan(y/x)

Where r, θ, and φ represent the spherical coordinates and x, y, and z represent the Cartesian coordinates.

## 4) What are the applications of transforming Cartesian coordinates into spherical harmonics?

Transforming Cartesian coordinates into spherical harmonics is useful in many fields, including physics, engineering, and computer graphics. It allows for a more efficient and accurate representation of three-dimensional objects, making it easier to analyze and manipulate them.

## 5) Are there any limitations to transforming Cartesian coordinates into spherical harmonics?

While transforming Cartesian coordinates into spherical harmonics is a useful tool, it is not always necessary or appropriate. In some cases, using Cartesian coordinates may be simpler and more practical. Additionally, the equations for transforming between the two coordinate systems can become more complex for higher dimensions.

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