What are "Spherical harmonics"?

Background: To undestand the equivarance of a spherical harmonics on rotation

A spherical harmonics is a function defined on the surface of a sphere that assigns a value to each point on the sphere. Mathematically, if we denote the sphere by S^2, then a function f defined on the sphere is written as f: S^2→C (or \mathbb{R} if the function is real-valued). This means that for every point on the sphere, specified by its coordinates (θ,ϕ), the function f assigns a complex (or real) number.

Coordinates on the Sphere

To specify points on the surface of a sphere, we use spherical coordinates:

• θ: Polar angle (colatitude), ranging from 0 to π.
• ϕ: Azimuthal angle (longitude), ranging from 0 to 2π.

What are 'l' and 'm' doing?

In summary, ℓ and m are essential in defining the spherical harmonics' complexity and symmetry.

The Degree (ℓ)

The degree ℓ is a non-negative integer (ℓ=0,1,2,…) that specifies the overall complexity and number of nodal lines (where the function value is zero) of the spherical harmonic. It determines:

1. The number of nodal lines: These are lines (or circles) on the sphere where the harmonic function is zero. Higher ℓ values correspond to more nodal lines.
2. The angular frequency: The degree ℓ represents the total number of peaks and troughs (oscillations) that occur on the sphere. It influences the spatial frequency of the pattern on the sphere.
3. The number of distinct spherical harmonics: For a given ℓ, there are 2ℓ+1distinct spherical harmonics, corresponding to different values of mmm.

The Order (m)

The order m is an integer that ranges from −ℓ to ℓ (i.e., m=−ℓ,−ℓ+1,…,ℓ−1,ℓ). It determines:

1. Azimuthal variation: The order m specifies the number of oscillations (or how the function varies) in the azimuthal direction (around the z-axis). It determines the dependence on the azimuthal angle ϕ.
2. Symmetry: The order m affects the symmetry of the function. For example, harmonics with m=0 are zonal harmonics (symmetric about the z-axis), while those with m=±ℓ are sectorial harmonics (having no symmetry about the z-axis).
3. Phase: The sign of m can be associated with the direction of rotation in the complex exponential term e^{imϕ}.

Examples

Real (Laplace) spherical harmonics 𝑌ℓ𝑚  for ℓ=0,…,4  (top to bottom) and 𝑚=0,…,ℓ  (left to right). Zonal, sectoral, and tesseral harmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonics 𝑌ℓ(−𝑚)  would be shown rotated about the z axis by 90∘/𝑚  with respect to the positive order ones.) - https://en.wikipedia.org/wiki/Spherical_harmonics

The above visualization is visualizing the spherical harmonics as heat maps or color-coded surfaces where different colors represent different values of the function.

• Y_0^0​: A uniform color across the sphere. (Monopole)
• Y_1^0​: One hemisphere is one color, the other hemisphere a different color. (Dipole)
• Y_2^0​: Three bands of color, alternating in sign (positive, negative, positive). (Quadrupole)
• Y_3^0​: Four bands of color.

Mathematical Expression

What is this visualization?

https://e3nn.org/

The other common visualization of spherical harmonics that doesn't involve mapping onto the surface of a sphere is a three-dimensional plot of the radial function r⋅∣Yℓm(θ,ϕ)∣. This method of visualization highlights the angular dependence of the spherical harmonics in a 3D space by extending the values radially from the origin.

 

내가 알고 있던 visualization은 graphics에서 쓰이고, 이거는 quantun mechanics, acoustics, electromagnetic theory에 쓰임.