Coefficient spherical harmonics

In summary, the conversation is about the calculation of ##Y_{\ell}^m## for a specific value of ##\ell## and ##m##. The conversation also discusses discrepancies between the solution provided by Mathematica and the one obtained using a different definition of the associated Legendre functions. This difference may be due to varying conventions and definitions used in special functions theory.
  • #1
Dustinsfl
2,281
5
$$
Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi}
$$

For ##\ell = m = 1##, we have
$$
\sqrt{\frac{(2 + 1)(0)!}{4\pi(2)!}}P^1_{1}(\cos\theta)e^{i\varphi} = \frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\varphi}\sin \theta
$$

But Mathematica is telling me the solution is
$$
-\frac{1}{2} e^{i\varphi} \sqrt{\frac{3}{2\pi}} \sin\theta
$$

What is going wrong?
 
Physics news on Phys.org
  • #2
Have you checked the definitions/conventions for the associated Legendre functions ? The definitions for Mathematica you can find on the functions.wolfram.com website. Unfortunately, it's difficult to say that special functions theory is a unitary process with unique definitions.
 
  • #3
dextercioby said:
Have you checked the definitions/conventions for the associated Legendre functions ? The definitions for Mathematica you can find on the functions.wolfram.com website. Unfortunately, it's difficult to say that special functions theory is a unitary process with unique definitions.

Some are defined with a (-1)^m which is weird my professor was using that definition since we were not in class.
 

1. What are coefficient spherical harmonics?

Coefficient spherical harmonics are mathematical functions used to represent the spherical symmetry of physical systems. They are a set of complex-valued functions that describe the angular dependence of a spherical harmonic function.

2. How are coefficient spherical harmonics used in science?

Coefficient spherical harmonics are used in various fields of science, such as physics, chemistry, and mathematics. They are particularly useful in solving problems related to spherical systems, such as the motion of celestial bodies or the behavior of atoms and molecules.

3. What is the significance of the coefficients in coefficient spherical harmonics?

The coefficients in coefficient spherical harmonics represent the amplitudes of the different harmonics in the expansion of a spherical function. They provide information about the strength and direction of the spherical symmetry in a physical system.

4. How are coefficient spherical harmonics calculated?

The coefficients in coefficient spherical harmonics can be calculated using various methods, such as numerical integration or analytical techniques. These calculations involve solving a system of equations to determine the coefficients that best fit the spherical function being studied.

5. What are some applications of coefficient spherical harmonics?

Coefficient spherical harmonics have many applications in science and engineering. They are commonly used in the analysis of data from satellite missions, such as mapping of Earth's gravity field. They are also used in computer graphics to generate realistic images of 3D objects with spherical symmetry.

Similar threads

  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
757
  • Advanced Physics Homework Help
Replies
1
Views
799
  • Advanced Physics Homework Help
Replies
5
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
874
  • Advanced Physics Homework Help
Replies
4
Views
268
  • Advanced Physics Homework Help
Replies
2
Views
747
  • Advanced Physics Homework Help
Replies
10
Views
2K
  • Advanced Physics Homework Help
Replies
6
Views
1K
  • Advanced Physics Homework Help
Replies
13
Views
2K
Back
Top