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^, then a function 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 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:
- 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.
- 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.
- The number of distinct spherical harmonics: For a given ℓ, there are 2ℓ+1distinct spherical harmonics, corresponding to different values of m.
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:
- 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.
- 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).
- Phase: The sign of can be associated with the direction of rotation in the complex exponential term e^{i.
Examples
The above visualization is visualizing the spherical harmonics as heat maps or color-coded surfaces where different colors represent different values of the function.
- : 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?
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.