Undergrad Transforming Cartesian Coordinates in terms of Spherical Harmonics

[Athenian]

TL;DR

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!

 

[eys_physics]

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.